Part 2 of 3 The function $$ f(x)=2 x+5 $$ is one-to-one. a. Find an equation for $$ f^{-1} $$, the inverse function. b. Verify that your equation is correct by showing that $$ f\left(f^{-1}(x)\right)=x $$ and $$ f^{-1}(f(x))=x $$. (OMipmy y A. $$ f^{-1}(x)= $$ IIII, for $$ x \neq $$ IIII B. $$ f^{-1}(x)=\frac{x-5}{2} $$, for all $$ x $$ C. $$ f^{-1}(x)=\ldots $$, for $$ x \leq $$ D. $$ f^{-1}(x)=\cdots $$, for $$ x \geq $$ b. Verify that the equation is correct. $$ \begin{aligned} f\left(f^{-1}(x)\right) & =f\left(\frac{x-5}{7}\right) & \text { and } & f^{-1}(f(x)) \end{aligned}=f^{-1}\left(\left(\frac{(x-5))}{}\right)\right. \text { Substitute. } $$

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UpStudy AI · Accepted Answer

Let $$ y = f(x) $$. Then
$$
y = 2x + 5.
$$
To find $$ f^{-1}(x) $$, we swap $$ x $$ and $$ y $$:
$$
x = 2y + 5.
$$
Now, solve for $$ y $$ by subtracting 5 from both sides:
$$
x - 5 = 2y.
$$
Then divide by 2:
$$
y = \frac{x-5}{2}.
$$
Thus, the inverse function is:
$$
f^{-1}(x)=\frac{x-5}{2}\quad\text{for all } x.
$$

To verify the inverse, we check both compositions:

1. Compute $$ f\left(f^{-1}(x)\right) $$:
$$
f\left(f^{-1}(x)\right) = f\left(\frac{x-5}{2}\right) = 2\left(\frac{x-5}{2}\right) + 5 = (x-5) + 5 = x.
$$

2. Compute $$ f^{-1}(f(x)) $$:
$$
f^{-1}(f(x)) = f^{-1}(2x+5) = \frac{(2x+5)-5}{2} = \frac{2x}{2} = x.
$$

Since both compositions return $$ x $$, the inverse is verified.
The inverse function is $$ f^{-1}(x) = \frac{x - 5}{2} $$ for all $$ x $$. To verify: $$ f\left(f^{-1}(x)\right) = x \quad \text{and} \quad f^{-1}(f(x)) = x $$ Both compositions confirm that the inverse is correct.

Part 2 of 3

The function is one-to-one.
a. Find an equation for , the inverse function.
b. Verify that your equation is correct by showing that and .
(OMipmy y
A. IIII, for IIII
B. , for all
C. , for
D. , for
b. Verify that the equation is correct.

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Part 2 of 3 The function is one-to-one. a. Find an equation for , the inverse function. b. Verify that your equation is correct by showing that and . (OMipmy y A. IIII, for IIII B. , for all C. , for D. , for b. Verify that the equation is correct.