If f(x)=-9x+9, what is the value of f-1(0)?

To find \( f^{-1}(0) \), we need to determine the inverse of the function \( f(x) = -9x + 9 \) and then evaluate it at 0.

### Step 1: Find the inverse function \( f^{-1}(x) \)

1. Start with the equation for \( f(x) \):
   \[
   y = -9x + 9
   \]
2. Swap \( x \) and \( y \) to begin finding the inverse:
   \[
   x = -9y + 9
   \]
3. Solve for \( y \):
   \[
   x - 9 = -9y
   \]
   \[
   y = \frac{9 - x}{9}
   \]
Thus, the inverse function is:
\[
f^{-1}(x) = \frac{9 - x}{9}
\]

### Step 2: Evaluate \( f^{-1}(0) \)

Now, substitute \( x = 0 \) into the inverse function:
\[
f^{-1}(0) = \frac{9 - 0}{9} = \frac{9}{9} = 1
\]

Therefore, the value of \( f^{-1}(0) \) is \( 1 \).

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Would you like further details or have any questions?

Here are 5 related questions:
1. How do you verify if two functions are inverses of each other?
2. What are the steps to find the inverse of a more complex function?
3. How can we graph the function \( f(x) = -9x + 9 \) and its inverse?
4. What happens to the inverse if the original function is not linear?
5. How do you determine if a function is invertible?

**Tip:** A function is invertible if it is one-to-one, meaning each output corresponds to exactly one input.

If f(x)=-9x+9, what is the value of f-1(0)?

Discover how to find the inverse of the linear function f(x) = -9x + 9 and evaluate it at f^{-1}(0). This detailed solution walks through the steps to solve inverse functions, focusing on core algebraic concepts for students in Grades 9-11.

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