Evaluating Functions
Given the functions \(f(x) = 3x - 7\) and \(g(x) = 2x^2 + 1\), find the value of \(f(g(2))\).
Arithmetic of Functions
Suppose that the functions $f$ and $g$ are defined as follows.
\[
f(x)=\frac{1}{x+5} \quad g(x)=\frac{8}{x}
\]
Find $\frac{f}{g}$. Then, give its domain using an interval or union of intervals. Simplify your answers.
\[
\left(\frac{f}{g}\right)(x)=
\]
Domain of $\frac{f}{g}$ :
Domain of Composite Functions
If we have two functions, \(f(x)=\sqrt{x}\) and \(g(x)=2x+3\), what is the domain of the composite function \(f(g(x))\)?
Finding the Sum
Given two functions: \(f(x) = 3x^2 + 2x + 1\) and \(g(x) = 2x^2 - x + 3\), what is \(f(x) + g(x)\)?
Finding the Difference
Given the functions \( f(x) = 2x - 3 \) and \( g(x) = x^2 - 5 \), find \( (f - g)(x) \)
Finding the Product
Given the functions \(f(x) = 4x + 3\) and \(g(x) = 2x^2 - 5\), find the product \((f \cdot g)(x)\).
Finding the Quotient
Given the functions \(f(x) = 3x^2 - 2x + 1\) and \(g(x) = x - 1\), find the quotient \(\frac{f(x)}{g(x)}\).
Finding the Domain of the Sum of the Functions
Given the 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 \(f(x) = \sqrt{x - 1}\) and \(g(x) = \frac{1}{x + 2}\).
Finding the Domain of the Product of the Functions
Given the functions \(f(x) = \frac{1}{x+2}\) and \(g(x) = x^2 - 4\), find the domain of the product of the functions \(h(x) = f(x)g(x)\).
Finding the Domain of the Quotient of the Functions
Given two functions \(f(x) = x^2 - 4\) and \(g(x) = 2x - 4\), find the domain of the quotient function \(h(x) = \frac{f(x)}{g(x)}\).
Finding Roots (Zeros)
Given two functions \( f(x) = x^2 - 3x + 2 \) and \( g(x) = -x^2 + 5x - 6 \). Find the roots of the function \( h(x) = f(x) - g(x) \).
Identifying Zeros and Their Multiplicities
Given the function f(x) = (x - 2)^2 (x + 3)^3, identify the zeros of the function and their multiplicities.
Finding the Inverse
Given the function \(f(x) = 2x + 3\), find the inverse function \(f^{-1}(x)\).
Finding Maximum Number of Real Roots
Given a polynomial function of degree 5, \(f(x) = 2x^5 - 3x^4 + 2x^3 - x^2 + 3x - 2\), find the maximum number of real roots that this function can have.
Function Composition
For $f(x)=\frac{4}{x+4}$ and $g(x)=\frac{3}{x}$, find
a. $(f \circ g)(x)$;
b. the domain of $f \circ g$
a. $(f \circ g)(x)=\frac{4 x}{3+4 x}$
(Simplify your answer.)
b. What is the domain of $f \circ g$ ?
The domain is
(Simplify your answer. Type your answer in interval notation. the expression.)
Finding the Domain and Range
Consider the function \(f(x) = \sqrt{x} \) and \(g(x) = x^2 \). Find the domain and range of \((f+g)(x)\), \((f-g)(x)\), \((fg)(x)\), and \((f/g)(x)\).