PCAT Exam: Quantitative Reasoning - A Simple Guide To Precalculus

Functions

Graphing and Identifying Domains, Ranges, Intercepts, and Zeros of Exponential Functions
The logarithmic function with base is denoted

Its base must be greater than 0 and not equal to 1, and the domain is all

The exponential function with base is denoted

Exponential and logarithmic functions with base are inverses.

By definition, if

Because exponential and logarithmic functions are inverses, the graph of one is obtained by reflecting the other over the line = .

A common base used is and in this case and its inverse is commonly written as the natural logarithmic function

 

Here is the graph of both functions:

 
The -intercept of the logarithmic function with any base is always the ordered pair

By the definition of inverse, the point always lies on the exponential function

This is true because any real number raised to the power of 0 equals 1.

Therefore, the exponential function only has a -intercept.

The exponential function also has a horizontal asymptote of the -axis as approaches negative infinity.

Because the graph is reflected over the line , to obtain the graph of the logarithmic function, the asymptote is also reflected. Therefore, the logarithmic function has a one-sided vertical asymptote at .

These asymptotes can be seen in the above graphs of and   

Logarithms
When working with logarithms and exponential expressions, it is important to remember the relationship between the two.

In general, the logarithmic form is for an exponential form .

Logarithms and exponential functions are inverses of each other.

A logarithmic scale is a scale of measurement that uses the logarithm of the given units instead of the actual given units. Each tick mark on such a scale is the product of the previous tick mark multiplied by a number. The advantage of using such a scale is that if one is working with large measurements, this technique reduces the scale into manageable quantities that are easier to read.

The Richter magnitude scale is the famous logarithmic scale used to measure the intensity of earthquakes, and the decibel scale is commonly used to measure sound level in electronics.

Solving Problems Related to Exponential and Logarithmic Functions
To solve an equation involving exponential expressions, the goal is to isolate the exponential expression. Once this process is completed, the logarithm—with the base equaling the base of the exponent of both sides—needs to be taken to get an expression for the variable.

If the base is e, the natural log of both sides needs to be taken.
To solve an equation with logarithms, the given equation needs to be written in exponential form, using the fact that means and then solved for the given variable.

Lastly, properties of logarithms can be used to simplify more than one logarithmic expression into one.

Some equations involving exponential and logarithmic functions can be solved algebraically, or analytically.

To solve an equation involving exponential functions, the goal is to isolate the exponential expression.

Then, the logarithm of both sides is found in order to yield an expression for the variable.

Laws of Logarithms will be helpful at this point.
To solve an equation with logarithms, the equation needs to be rewritten in exponential form.

The definition that means needs to be used.

Then, one needs to solve for the given variable.

Properties of logarithms can be used to simplify multiple logarithmic expressions into one.

Other methods can be used to solve equations containing logarithmic and exponential functions. Graphs and graphing calculators can be used to see points of intersection. In a similar manner, tables can be used to find points of intersection. Also, numerical methods can be utilized to find approximate solutions.

Exponential Growth and Decay
Exponential growth and decay are important concepts in modeling real-world phenomena.

The growth and decay formula is

where the independent variable represents temperature, represents an initial quantity, represents the rate of increase or decrease, and represents the amount of the quantity at time

If the equation models exponential growth and a common application is population growth. If the equation models exponential decay and a common application is radioactive decay.

Exponential and logarithmic solving techniques are necessary to work with the growth and decay formula.

Using Exponential and Logarithmic Functions in Finance Problems
Modeling within finance also involves exponential and logarithmic functions.

Compound interest results when the bank pays interest on the original amount of money – the principal – and the interest that has accrued.

The compound interest equation is

where is the principal, is the interest rate, is the number of times per year the interest is compounded, and is the time in years. 

The result, is the final amount after years.

Mathematical problems of this type that are frequently encountered involve receiving all but one of these quantities and solving for the missing quantity.

The solving process then involves employing properties of logarithmic and exponential functions.

Interest can also be compounded continuously.

This formula is given as

If $1,000 was compounded continuously at a rate of 2% for 4 years, the result would be

Rate of Change Proportional to the Current Quantity
Many quantities grow or decay as fast as exponential functions.

Specifically, if such a quantity grows or decays at a rate proportional to the quantity itself, it shows exponential behavior.

If a data set is given with such specific characteristics, the initial amount and an amount at a specific time, can be plugged into the exponential function for and .

Using properties of exponents and logarithms, one can then solve for the rate, .

This solution yields enough information to have the entire model, which can allow for an estimation of the quantity at any time, and the ability to solve various problems using that model.

Graphing and Identifying Domains, Ranges, Intercepts, Zeros, and Inverses of the Circular Functions
From the unit circle, the trigonometric ratios were found for the special right triangle with a hypotenuse of 1.


From this triangle, the following Pythagorean identities are formed:

, , and .

The second two identities are formed by manipulating the first identity.

Since identities are statements that are true for any value of the variable, then they may be used to manipulate equations.

For example, a problem may ask for simplification of the expression

.

Using the fact that ,  can then be substituted in for , making the expression .

Then the two terms on top and bottom cancel each other out, simplifying the expression to .

By the first Pythagorean identity stated above, the expression can be turned into .

Another set of trigonometric identities are the double-angle formulas:


Using these formulas, the following identity can be proved: .

By using one of the Pythagorean identities, the denominator can be rewritten as .

By knowing the reciprocals of the trigonometric identities, the secant term can be rewritten to form the equation .

Replacing , the equation becomes , where the can cancel out.

The new equation is

.

 

This final equation is one of the double-angle formulas.
Other trigonometric identities such as half-angle formulas, sum and difference formulas, and difference of angles formulas can be used to prove and rewrite trigonometric equations.

Depending on the given equation or expression, the correct identities need to be chosen to write equivalent statements.
The graph of sine is equal to the graph of cosine, shifted units.

Therefore, the function is equal to .

Within functions, adding a constant to the independent variable shifts the graph either left or right.

By shifting the cosine graph, the curve lies on top of the sine function. By transforming the function, the two equations give the same output for any given input.

Solving Trigonometric Functions
Solving trigonometric functions can be done with a knowledge of the unit circle and the trigonometric identities. It requires the use of opposite operations combined with trigonometric ratios for special triangles.

For example, the problem may require solving the equation for the values of between 0 and 180 degrees.

The first step is to factor out the term, resulting in .

By the factoring method of solving, each factor can be set equal to zero: and .

The second equation can be solved to yield the following equation:

.

Now that the value of is found, the trigonometric ratios can be used to find the solutions of .

Solving trigonometric functions requires the use of algebra to isolate the variable and a knowledge of trigonometric ratios to find the value of the variable.

The unit circle can be used to find answers for special triangles.

Beyond those triangles, a calculator can be used to solve for variables within the trigonometric functions.

Performing Algebraic Operations on Functions
When two functions are added together, the result is known as a sum function.

The domain of a sum function is the intersection of the two domains of the original functions.

If the functions are and , then it is possible to compute an expression for by adding together those functions.

For example, suppose that and .

Then the expression for .

In this example, the domain of both and is all real numbers, so the domain of the sum function listed above is all real numbers.

When one function is subtracted from another, the result is known as a difference function.

Like the sum function, the domain of a difference function is also the intersection of the two domains of the original functions.

With the functions from the previous paragraph, the subtraction is .

The domain of the difference function listed above is also all real numbers.

When finding an expression for such subtraction of functions, it is very important to make sure to subtract the entire second function from the first function.
The result of the multiplication of two functions is known as a product function.

Like sum and difference formulas, the domain of the product function is the intersection of the two domains of the original functions.

In this example, the domain of also written as is all real numbers.
Finally, the quotient function is found by the division of two functions.

By definition, .

With the above examples, .

Note that the denominator cannot be zero in a fraction, so provided that . Therefore, that the domain of is the intersection of the domains of both and , excluding any -values that cause

In the example discussed previously, both 2 and 0 would create a 0 denominator when plugged in for in the function.

Therefore, the domain of the quotient function in interval notation is
The difference quotient is an important calculation and it involves performing algebraic operations on functions.

For a function the difference quotient is defined as

Therefore, for the function its difference quotient is

Identifying and Using Composite Functions
In many scenarios, it is common for the output of one function to depend on an output of another function. Therefore, the output of the second function would be the input of the first function.

A real-life situation can be seen with taxes. If you work hourly, your paycheck is based on the number of hours worked, and the taxes are based on the total paycheck.

This involves what is called a composite function.
Given two functions f and g, the composition of g with f is written , and it is defined by the equation .

In other words, first apply the function f to , and then apply g to the result.
The domain of such a function is some subset of the domain of f.

Specifically, it is those values in the domain of f, such that the result when f yields something in the domain of g.

Another way to think about a composite function is that the function gets plugged into whenever there is an

Note that is in the domain of and is in the domain of

Also, the domain of a composite function can be thought of the intersection of the domain of the input function and the domain of the composite function.

More than two functions can be involved when building composite functions as well.

Consider

and

The composite function

Its domain is the intersection of the two input functions and the composite function, which is all real numbers.

Complex Numbers
Given a complex number its complex conjugate is

For example, the complex conjugate of is

It is the number with equal real part and opposite imaginary part.

The absolute value of a complex number also known as its magnitude or modulus is

It is equal to the distance from the origin to the corresponding point in the complex plane.

A multiplicative inverse of any number is what one multiplies by to obtain a product of 1.

In other words, it is the reciprocal. For a complex number its multiplicative inverse is which can be rewritten once rationalized as

The complex numbers form a field, which means that two complex numbers can be added together and multiplied times one another and either result is still a complex number.

Also, for any complex number its additive inverse is also a complex number.

Finally, every nonzero complex number has a multiplicative inverse that is a complex number.

Addition can be performed using the same process as vector addition.

In this case, addition is done component-wise. In component form, , in which the real part and imaginary part are considered separate components.

Representing Complex Numbers
Given the situation, the format in which a complex number is used is important. When given as it is written in its algebraic, rectangular, or Cartesian form, and it relates to an ordered pair in the complex plane with a real (horizontal) and imaginary (vertical) axis.

A complex number that is completely imaginary lies on the vertical axis.

Its vector form is found similarly by writing its real and imaginary parts into vector form

Polar form of a complex number is necessary if there is desire to use complex numbers in the real number system by using polar coordinates.

In this case,

where represents the absolute value of and

The ordered pair represent the polar coordinates.

Finally, a complex number can be represented in exponential form, and Euler’s formula is necessary.

With this formula, the complex number’s exponential form is

Also, it is true that and

Complex Number Operations
Complex number operations can be performed using geometric representations of the numbers themselves. In terms of addition, two complex numbers can be added by adding the real parts together separately from the imaginary parts.

This operation can be performed within the ordered pairs.

Multiplication can be thought of using polar coordinates.

Its magnitude , which is the distance from the point to the origin in the complex plane, and its argument which is the angle from the horizontal axis to the line segment connecting the origin and the point itself, can be used.

Two complex numbers can be multiplied by one another by multiplying their magnitudes together and adding their arguments together. Therefore, the product of the complex number with magnitude and argument with the complex number with magnitude and argument is the complex number with magnitude and argument .

Complex Numbers as Solutions
Complex numbers may result from solving polynomial equations using the quadratic equation. Since complex numbers result from taking the square root of a negative number, the number found under the radical in the quadratic formula—called the determinant—tells whether or not the answer will be real or complex. If the determinant is negative, the roots are complex.

Even though the coefficients of the polynomial may be real numbers, the roots are complex.
Solving polynomials by factoring is an alternative to using the quadratic formula.

For example, in order to solve for x, it needs to be factored. It factors into

The solution set can be found by setting each factor equal to zero, resulting in .

When is negative, the factors are complex numbers.

For example, can be factored into .

The two roots are then found to be .

When dealing with polynomials and solving polynomial equations, it is important to remember the fundamental theorem of algebra.

When given a polynomial with a degree of n, the theorem states that there will be n roots.

These roots may or may not be complex. For example, the following polynomial equation of degree 2 has two complex roots: .

The factors of this polynomial are and , resulting in the roots .

As seen on the graph below, imaginary roots occur when the graph does not touch the x-axis.


When a graphing calculator is permitted, the graph can always confirm the number and types of roots of the polynomial.
A polynomial identity is a true equation involving polynomials.

For example, which can be proved through multiplication by the FOIL method and factoring.

This idea can be extended to involve complex numbers.

Because

This identity can also be proven through FOIL and factoring.

Polar Representations of Complex Numbers
A complex number in the form  can be written in its polar form if there is necessity to use it amongst real numbers in the Cartesian coordinate system.

In this case, where represents the absolute value of , the distance from the point to the origin, and represents the angle, in radians, from the positive x-axis to the ray that connects the origin to the point.

The ordered pair represents the polar coordinates.

Given the polar representation, a proof by induction can be used to obtain DeMoivre’s Theorem, which says that if n is a natural number,

Vectors
A vector can be thought of as a list of numbers. These can be thought of as an abstract list of numbers, or else as giving a location in a space. For example, the coordinates for points in the Cartesian plane are vectors.

Each entry in a vector can be referred to by its location in the list: first, second, and so on.

The total length of the list is the dimension of the vector.

A vector is often denoted as such by putting an arrow on top of it,

e.g.

Adding Vectors Graphically and Algebraically
There are two basic operations for vectors.

First, two vectors can be added together.

Let

.

The the sum of the two vectors is defined to be

.

Subtraction of vectors can be defined similarly.
Vector addition can be visualized in the following manner. First, visualize each vector as an arrow. Then place the base of one arrow at the tip of the other arrow. The tip of this first arrow now hits some point in the space, and there will be an arrow from the origin to this point. This new arrow corresponds to the new vector. In subtraction, we reverse the direction of the arrow being subtracted.
For example, consider adding together the vectors (-2, 3) and (4, 1). The new vector will be (-2+4, 3+1), or (2, 4). Graphically, this may be pictured in the following manner.

Performing Scalar Multiplications
The second basic operation for vectors is called scalar multiplication.

Scalar multiplication allows us to multiply any vector by any real number, which is denoted here as a scalar. Let , and let a be an arbitrary real number.

Then the scalar multiple

Graphically, this corresponds to changing the length of the arrow corresponding to the vector by a factor, or scale, of a.

That is why the real number is called a scalar in this instance.
As an example, let .

Then .
Note that scalar multiplication is distributive over vector addition, meaning that .

Representing Vectors Equations of Lines and Planes
Since vectors can be thought of as giving directions, and since lines continue on in a single direction, it is possible to represent any line by using vectors.

To do so requires two things: a vector that goes to a point on the line, and a vector which gives the direction of a line.

The equation for the line will then be all vectors of the form , where s can take the value of any real number.
Suppose we know two points on the line, A and B.

Then we can take to be the vector pointing to A, and take to be the vector that goes from A to B.

This will be the vector going to B minus the vector going to A. Of course, there will be many different vector equations corresponding to the same line, since any two points on the line may be used.

Consider a line in the Cartesian plane which passes through the points (1, -2) and (2, 3).

Call the first point A and the second point B.

Then we can take , and .

Then the vector equation for the line will be .

A plane in three dimensions can similarly be represented by using vectors.

In this case, three vectors are needed: first, a vector pointing to some point on the plane, and then two vectors and corresponding to the two directions in which the plane goes.

If three points on the plane are given, A, B, and C, then one can take to be the vector pointing to A, to be the vector from A to B, and to be the vector from A to C.

The vector equation for the plane is then .

Note, however, that this requires the three given points to not all lie on the same line.

If they all lie upon a single line, then they do not define a unique plane.

Suppose, then, that the points lie on a plane.

We can take , , and .

 

The vector equation for the plane will now be .
Two vectors are equal if, and only if, their individual components are equal.

For example, but

Also, vector addition is performed component-wise.

Such an example is

The zero vector is defined to be the vector containing only components equal to 0, and any vector plus a zero vector equals itself.

Hence, the zero vector is the additive identity. Scalar multiplication is performed by multiplying each component by the scalar.

For example,

Scalar multiplication and addition can be used to prove that the distributive property holds within vector addition and scalar addition. Vector multiplication is defined using the dot product, which is also known as the scalar product.

The result of a dot product is a scalar.

Each corresponding component is multiplied, and then the sum of all products is found. For example, Alternatively, the dot product is defined to be the product of the magnitudes of each vector and the cosine of the angle between the two vectors.

Therefore, if two vectors are perpendicular, their dot product is equal to zero.

Finally, two vectors are parallel if they are scalar multiples of each other.