Find the inverse of f(x)=x^(2)-5,x <= 0

Write Function Swap: Write down the function and swap x and y to begin finding the inverse. We have f(x) = x^2 - 5 for x ≤ 0. To find the inverse, we replace f(x) with y and then swap x and y to solve for the new y. y = x^2 - 5 becomes x = y^2 - 5.Solve for Inverse: Solve for y to find the inverse function. We need to isolate y on one side of the equation. Start by adding 5 to both sides of the equation. x + 5 = y^2.Square Root Solve: Take the square root of both sides to solve for y. Since we are looking for the inverse function where the original function is defined for x ≤ 0, we need to take the negative square root because the output of the original function is non-positive when x is non-positive. y = -√(x + 5).Replace with Notation: Replace y with the inverse notation. The inverse function is denoted as f^(-1)(x), so we replace y with this notation. f^(-1)(x) = -√(x + 5).State Domain: State the domain of the inverse function. The domain of the inverse function is the range of the original function. Since the original function is defined for x ≤ 0 and is a downward-opening parabola (because we only consider x ≤ 0), the range is all y ≤ 6 (the y-coordinate of the vertex of the parabola). Therefore, the domain of the inverse function is x ≤ 6.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the inverse of $\newline$ $f(x)=x^{2}-5,\,x \leq 0$

View step-by-step help

Home

Math Problems

Algebra 2

Find the vertex of the transformed function

Full solution

Q. Find the inverse of

\newline

f(x)=x^{2}-5,\,x \leq 0

Write Function Swap: Write down the function and swap $x$ and $y$ to begin finding the inverse. $\newline$ We have $f(x) = x^2 - 5$ for $x \leq 0$ . To find the inverse, we replace $f(x)$ with $y$ and then swap $x$ and $y$ to solve for the new $y$ . $\newline$ $y = x^2 - 5$ becomes $y$ $0$ .
Solve for Inverse: Solve for $y$ to find the inverse function. $\newline$ We need to isolate $y$ on one side of the equation. Start by adding $5$ to both sides of the equation. $\newline$ $x + 5 = y^2$ .
Square Root Solve: Take the square root of both sides to solve for $y$ . $\newline$ Since we are looking for the inverse function where the original function is defined for $x \leq 0$ , we need to take the negative square root because the output of the original function is non-positive when $x$ is non-positive. $\newline$ $y = -\sqrt{x + 5}$ .
Replace with Notation: Replace $y$ with the inverse notation. $\newline$ The inverse function is denoted as $f^{-1}(x)$ , so we replace $y$ with this notation. $\newline$ $f^{-1}(x) = -\sqrt{x + 5}$ .
State Domain: State the domain of the inverse function. $\newline$ The domain of the inverse function is the range of the original function. Since the original function is defined for $x \leq 0$ and is a downward-opening parabola (because we only consider $x \leq 0$ ), the range is all $y \leq 6$ (the $y$ -coordinate of the vertex of the parabola). Therefore, the domain of the inverse function is $x \leq 6$ .