25. (M0) Use Structure Write an equation for the inverse of the general linear function $$ f(x)=a x+b $$ where $$ a
eq 0 $$. Then use your equation to find the inverse of $$ f(x)=7 x-2 $$.

Question

25. (M0) Use Structure Write an equation for the inverse of the general linear function $$ f(x)=a x+b $$ where $$ a 
eq 0 $$. Then use your equation to find the inverse of $$ f(x)=7 x-2 $$.

UpStudy AI · Accepted Answer

**Step 1. Express the function as an equation.**  
Let  
$$
y = ax + b.
$$

**Step 2. Swap $$ x $$ and $$ y $$.**  
We have  
$$
x = ay + b.
$$

**Step 3. Solve for $$ y $$.**  
Subtract $$ b $$ from both sides:  
$$
x - b = ay.
$$  
Then divide both sides by $$ a $$ (noting that $$ a \neq 0 $$):  
$$
y = \frac{x - b}{a}.
$$

Thus, the inverse function for the general linear function $$ f(x) = ax + b $$ is  
$$
f^{-1}(x) = \frac{x - b}{a}.
$$

**Step 4. Find the inverse of $$ f(x)=7x-2 $$.**  
For the function $$ f(x)=7x-2 $$, we have $$ a = 7 $$ and $$ b = -2 $$. Substitute these values into the formula for the inverse function:  
$$
f^{-1}(x) = \frac{x - (-2)}{7} = \frac{x + 2}{7}.
$$

Thus, the inverse of $$ f(x)=7x-2 $$ is  
$$
f^{-1}(x)=\frac{x+2}{7}.
$$
The inverse of the function $$ f(x) = 7x - 2 $$ is $$ f^{-1}(x) = \frac{x + 2}{7} $$.

25. (M0) Use Structure Write an equation for the inverse of the general linear function \( f(x)=a x+b \) where \( a \neq 0 \). Then use your equation to find the inverse of \( f(x)=7 x-2 \).

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