Let f(x) = \frac ( x + 3) ( x + 7 ), find f^-1(-7)

We are given the function:

\[
f(x) = \frac{x+3}{x+7}
\]

and we need to find \( f^{-1}(-7) \), which means finding the value of \( x \) such that:

\[
f(x) = -7
\]

So, we set \( f(x) = -7 \) and solve for \( x \):

\[
\frac{x+3}{x+7} = -7
\]

### Step 1: Multiply both sides by \( (x + 7) \) to eliminate the denominator:

\[
x + 3 = -7(x + 7)
\]

### Step 2: Expand the right-hand side:

\[
x + 3 = -7x - 49
\]

### Step 3: Move all the terms involving \( x \) to one side:

\[
x + 7x = -49 - 3
\]

\[
8x = -52
\]

### Step 4: Solve for \( x \):

\[
x = \frac{-52}{8} = -6.5
\]

Thus, \( f^{-1}(-7) = -6.5 \).

Would you like more details or have any questions?

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Here are some related questions for deeper understanding:

1. How do you derive the general formula for \( f^{-1}(x) \)?
2. What happens if the original function has no inverse?
3. How would you graph the function \( f(x) = \frac{x+3}{x+7} \)?
4. What is the domain and range of the original function \( f(x) \)?
5. How would you verify that \( f^{-1}(x) \) is indeed the inverse of \( f(x) \)?

**Tip:** Always check the domain restrictions when working with functions involving fractions.

Let f(x) = \frac{x+3}{x+7}, find f^-1(-7).

Discover how to find the inverse of the rational function f(x) = (x+3)/(x+7) by solving for f^-1(-7). This step-by-step guide covers important concepts like inverse functions, rational equations, and algebraic manipulation, perfect for high school students in Grades 10-12.

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