Question

homework 4.6

1. consider the function $f(x) = \frac{1}{2}x + 5$

a) graph the function.
b) find the equation of the inverse.
c) graph the inverse, $f^{-1}(x)$.
d) is the inverse a function?
why / why not?

fill in each table for the inverse relation. is the inverse a function?
2.

x	-2	-1	0	1	2	3

x

function?
3.

x	-2	1	0	1	2	3

x

function?
algebraically find the inverse of each function.

1. $f(x) = x + 3$
2. $f(x) = 4x - 1$
3. $f(x) = \sqrt{x + 2}$
4. $f(x) = \frac{x + 7}{3}$

Explanation:

Problem 1

Step1: Rewrite function with $y$

Let $y = f(x)$, so $y = \frac{1}{2}x + 5$

Step2: Swap $x$ and $y$ (inverse step)

$x = \frac{1}{2}y + 5$

Step3: Solve for $y$

Subtract 5: $x - 5 = \frac{1}{2}y$
Multiply by 2: $y = 2(x - 5) = 2x - 10$

Step4: Verify function status

The inverse is linear, passes vertical line test.

Step1: Swap $x$ and $y$ values

Take original $(x,y)$ pairs: $(-2,9), (-1,3), (0,-4), (1,8), (2,-6), (3,3)$; swap to get inverse pairs $(9,-2), (3,-1), (-4,0), (8,1), (-6,2), (3,3)$

Step2: Check function definition

Check if any $x$-value repeats with different $y$-values.

Step1: Swap $x$ and $y$ values

Take original $(x,y)$ pairs: $(-2,-7), (1,6), (0,8), (1,-1), (2,3), (3,7)$; swap to get inverse pairs $(-7,-2), (6,1), (8,0), (-1,1), (3,2), (7,3)$

Step2: Check function definition

Check if any $x$-value repeats with different $y$-values.

Answer:

a) To graph $f(x)=\frac{1}{2}x+5$: use y-intercept $(0,5)$ and slope $\frac{1}{2}$ (rise 1, run 2) to plot points and draw the line.
b) $f^{-1}(x) = 2x - 10$
c) To graph $f^{-1}(x)=2x-10$: use y-intercept $(0,-10)$ and slope $2$ (rise 2, run 1) to plot points and draw the line (it is the reflection of $f(x)$ over $y=x$).
d) Yes, the inverse is a function. It passes the vertical line test (each $x$-value maps to exactly one $y$-value).

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Problem 2