Questions: Solve the following inverse function: If (f(x)=7 x+2), find (f^-1(x)).

Transcript text: Solve the following inverse function: If $f(x)=7 x+2$, find $f^{-1}(x)$.

Solution

Solution Steps

Step 1: Given the linear function

Given the linear function $f(x) = 7x + 2$, we aim to find its inverse function $f^{-1}(x)$.

Step 2: Replace $f(x)$ with $y$

We replace $f(x)$ with $y$ to get $y = {a}x + {b}$. This is the first step in finding the inverse function.

Step 3: Swap $x$ and $y$

Next, we swap $x$ and $y$ to reflect the idea of finding an inverse function, resulting in $x = {a}y + {b}$.

Step 4: Solve for $y$

To express the inverse function in terms of $x$, we solve the equation for $y$. Subtracting $b$ from both sides gives us $x - {b} = {a}y$. Dividing by ${a}$ (assuming ${a} \neq 0$) to isolate $y$ gives us $y = \frac{{x - {b}}}{{{a}}}$.

Final Answer:

The inverse function $f^{-1}(x)$ is given by $f^{-1}(x) = \frac{x - 2}{7}$, rounded to 0 decimal places where necessary.

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