Find the Behavior (Leading Coefficient Test)
Given a polynomial function \(f(x) = -3x^5 + 2x^3 - 9x + 4\), what is the end behavior of the function?
Determining Odd and Even Functions
Determine whether the function \(f(x) = 3x^4 - 2x^2 + 1\) is an odd function, an even function, or neither.
Arithmetic of Functions
Consider the following functions.
\[
\begin{array}{l}
f=\{(-1,-2),(3,-1),(1,3)\} \\
\text { and } \\
g=\{(-1,-1),(3,0),(1,-3)\}
\end{array}
\]
and
Step 1 of $4:$ Find $(f+g)(1)$
Answer
How to enter your answer (opens in new window)
\[
(f+g)(1)=
\]
Finding Roots Using the Factor Theorem
Find the roots of the function \( f(x) = x^3 - 6x^2 + 11x - 6 \) using the Factor Theorem.
Finding All Possible Roots/Zeros (RRT)
Find all possible roots/zeros of the function \(f(x) = x^3 - 3x^2 - 4x + 12\) using Rational Root Theorem (RRT).
Determine if Surjective (Onto)
Determine whether the function \(f: \mathbb{Z} \to \mathbb{Z}\) defined by \(f(x) = 2x + 3\) is surjective (onto).
Finding the Vertex
Find the vertex of the function \(f(x) = -2x^2 + 4x + 5\).
Finding the Sum
Find the sum of the first 20 terms of the arithmetic sequence defined by the function \(f(n) = 3n+1\)
Finding the Difference
Let's consider two functions: \(f(x) = 3x + 2\) and \(g(x) = 5x - 1\). What is the difference between these two functions when \(x = 4\)?
Finding the Product
Find the product of the functions \(f(x) = 3x^2 + 2x + 1\) and \(g(x) = 2x^2 + 3x + 4\).
Finding the Quotient
Given two functions, \(f(x) = 2x + 3\) and \(g(x) = x + 1\), find the quotient \(\frac{f(x)}{g(x)}\).
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 \(f(x) + g(x)\).
Finding the Domain of the Difference of the Functions
Find the domain of the difference of the functions \(f(x) = \sqrt{x}\) and \(g(x) = \frac{1}{x}\).
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 \(f(x)g(x)\).
Finding the Domain of the Quotient of the Functions
Find the domain of the quotient of the functions \(f(x) = x^2 - 4\) and \(g(x) = x^2 - 1\).
Finding Roots (Zeros)
Find the roots of the function \(f(x) = 2x^2 - 5x - 3\).
Identifying Zeros and Their Multiplicities
Given the function \(f(x) = (x + 1)(x - 2)^2(x + 3)^3\), identify the zeros and their multiplicities.
Finding the Bounds of the Zeros
Given the function \(f(x) = 2x^{3} - 5x^{2} + 6x - 3\), find the bounds of the zeros of this function.
Proving a Root is on the Interval
Let's consider a function, \(f(x) = x^3 - 3x^2 - 4x + 12\). Prove that there is a root in the interval (-2, 0).
Finding the Average Rate of Change
Given a function \( f(x) = 2x^2 + 3x + 1 \), find the average rate of change of \( f \) on the interval \( [1, 4] \).
Finding the Inverse
Find the inverse of the function \( f(x) = 3x + 7 \)
Finding Maximum Number of Real Roots
Find the maximum number of real roots of the function \(f(x) = x^5 - 6x^4 + 9x^3 - 4x^2 + x - 2\).
Function Composition
Let \( f(x) = 3x + 2 \) and \( g(x) = x^2 - 5 \). Find the composition of the functions \( (f \circ g)(x) \) and \( (g \circ f)(x) \).
Rewriting as a Function
Let's suppose we are given a linear equation y = 2x + 5. Rewrite this equation as a function of x.
Finding the Domain and Range
Find the domain and range of the function \( f(x) = \sqrt{x - 2} \).
Determining if a Function is Rational
Determine if the function \( f(x) = \frac{x^2 + 2x + 1}{x^2 - 1} \) is a rational function.
Determining if a Function is Proper or Improper
Determine whether the following function is proper or improper: \(f(x) = \frac{x^{2} + 2x + 1}{x^{3} + x^{2} - x - 1}\)
Maximum/Minimum of Quadratic Functions
Given the quadratic function \(f(x) = 3x^2 - 12x + 7\), find the maximum or minimum value of the function.