Question

determine whether the function has an inverse function.
$f(x) = 4x + 5$
$\bigcirc$ yes, $f$ does have an inverse.
$\bigcirc$ no, $f$ does not have an inverse.
if it does, then find the inverse function. (if an answer does not exist, enter dne
$f^{-1}(x) = \square$

Explanation:

Step1: Check if function is one - to - one

A function \(y = f(x)\) has an inverse if and only if it is one - to - one (injective). The function \(f(x)=4x + 5\) is a linear function with a slope \(m = 4
eq0\). Linear functions with non - zero slopes are one - to - one because for any two distinct \(x_1\) and \(x_2\), if \(x_1
eq x_2\), then \(f(x_1)=4x_1 + 5\) and \(f(x_2)=4x_2+5\), and \(f(x_1)-f(x_2)=4(x_1 - x_2)
eq0\) (since \(x_1
eq x_2\)). So, \(f(x)\) is one - to - one and thus has an inverse.

Step2: Find the inverse function

1. Start with \(y = 4x+5\).
2. Swap \(x\) and \(y\): \(x = 4y + 5\).
3. Solve for \(y\):

• Subtract 5 from both sides: \(x - 5=4y\).
• Divide both sides by 4: \(y=\frac{x - 5}{4}=\frac{1}{4}x-\frac{5}{4}\).
• So, \(f^{-1}(x)=\frac{x - 5}{4}\).

Answer:

Yes, \(f\) does have an inverse. \(f^{-1}(x)=\frac{x - 5}{4}\)