Functions

Find the Behavior (Leading Coefficient Test)

Given a polynomial function

f (x) = - 3 x^{5} + 2 x^{3} - 9 x + 4

, what is the end behavior of the function?

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Determining Odd and Even Functions

Determine whether the function

f (x) = 3 x^{4} - 2 x^{2} + 1

is an odd function, an even function, or neither.

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Arithmetic of Functions

Consider the following functions.

\begin{array}{l} f = {(- 1, - 2), (3, - 1), (1, 3)} \\ and \\ g = {(- 1, - 1), (3, 0), (1, - 3)} \end{array}

and Step 1 of

4 :

Find

(f + g) (1)

Answer How to enter your answer (opens in new window)

(f + g) (1) =

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Finding Roots Using the Factor Theorem

Find the roots of the function

f (x) = x^{3} - 6 x^{2} + 11 x - 6

using the Factor Theorem.

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Finding All Possible Roots/Zeros (RRT)

Find all possible roots/zeros of the function

f (x) = x^{3} - 3 x^{2} - 4 x + 12

using Rational Root Theorem (RRT).

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Determine if Surjective (Onto)

Determine whether the function

f : Z \to Z

defined by

f (x) = 2 x + 3

is surjective (onto).

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Finding the Vertex

Find the vertex of the function

f (x) = - 2 x^{2} + 4 x + 5

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Finding the Sum

Find the sum of the first 20 terms of the arithmetic sequence defined by the function

f (n) = 3 n + 1

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Finding the Difference

Let's consider two functions:

f (x) = 3 x + 2

and

g (x) = 5 x - 1

. What is the difference between these two functions when

x = 4

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Finding the Product

Find the product of the functions

f (x) = 3 x^{2} + 2 x + 1

and

g (x) = 2 x^{2} + 3 x + 4

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Finding the Quotient

Given two functions,

f (x) = 2 x + 3

and

g (x) = x + 1

, find the quotient

\frac{f (x)}{g (x)}

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Finding the Domain of the Sum of the Functions

Given two functions,

f (x) = \sqrt{x - 2}

and

g (x) = \frac{1}{x + 3}

, find the domain of the sum of the functions

f (x) + g (x)

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Finding the Domain of the Difference of the Functions

Find the domain of the difference of the functions

f (x) = \sqrt{x}

and

g (x) = \frac{1}{x}

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Finding the Domain of the Product of the Functions

Let

f (x) = \sqrt{x + 2}

and

g (x) = \frac{1}{x}

. Find the domain of the product of the functions

f (x) g (x)

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Finding the Domain of the Quotient of the Functions

Find the domain of the quotient of the functions

f (x) = x^{2} - 4

and

g (x) = x^{2} - 1

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Finding Roots (Zeros)

Find the roots of the function

f (x) = 2 x^{2} - 5 x - 3

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Identifying Zeros and Their Multiplicities

Given the function

f (x) = (x + 1) (x - 2)^{2} (x + 3)^{3}

, identify the zeros and their multiplicities.

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Finding the Bounds of the Zeros

Given the function

f (x) = 2 x^{3} - 5 x^{2} + 6 x - 3

, find the bounds of the zeros of this function.

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Proving a Root is on the Interval

Let's consider a function,

f (x) = x^{3} - 3 x^{2} - 4 x + 12

. Prove that there is a root in the interval (-2, 0).

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Finding the Average Rate of Change

Given a function

f (x) = 2 x^{2} + 3 x + 1

, find the average rate of change of

f

on the interval

[1, 4]

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Finding the Inverse

Find the inverse of the function

f (x) = 3 x + 7

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Finding Maximum Number of Real Roots

Find the maximum number of real roots of the function

f (x) = x^{5} - 6 x^{4} + 9 x^{3} - 4 x^{2} + x - 2

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Function Composition

Let

f (x) = 3 x + 2

and

g (x) = x^{2} - 5

. Find the composition of the functions

(f \circ g) (x)

and

(g \circ f) (x)

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Rewriting as a Function

Let's suppose we are given a linear equation y = 2x + 5. Rewrite this equation as a function of x.

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Finding the Domain and Range

Find the domain and range of the function

f (x) = \sqrt{x - 2}

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Determining if a Function is Rational

Determine if the function

f (x) = \frac{x^{2} + 2 x + 1}{x^{2} - 1}

is a rational function.

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Determining if a Function is Proper or Improper

Determine whether the following function is proper or improper:

f (x) = \frac{x^{2} + 2 x + 1}{x^{3} + x^{2} - x - 1}

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Maximum/Minimum of Quadratic Functions

Given the quadratic function

f (x) = 3 x^{2} - 12 x + 7

, find the maximum or minimum value of the function.

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Find the Behavior (Leading Coefficient Test)

Determining Odd and Even Functions

Arithmetic of Functions

Finding Roots Using the Factor Theorem

Finding All Possible Roots/Zeros (RRT)

Determine if Surjective (Onto)

Finding the Vertex

Finding the Sum

Finding the Difference

Finding the Product

Finding the Quotient

Finding the Domain of the Sum of the Functions

Finding the Domain of the Difference of the Functions

Finding the Domain of the Product of the Functions

Finding the Domain of the Quotient of the Functions

Finding Roots (Zeros)

Identifying Zeros and Their Multiplicities

Finding the Bounds of the Zeros

Proving a Root is on the Interval

Finding the Average Rate of Change

Finding the Inverse

Finding Maximum Number of Real Roots

Function Composition

Rewriting as a Function

Finding the Domain and Range

Determining if a Function is Rational

Determining if a Function is Proper or Improper

Maximum/Minimum of Quadratic Functions

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