Find the inverse function of the function f, of, x, equals, minus, start fraction, 7, divided by, 8x, end fraction


To find the inverse function of \( f(x) = -\frac{7}{8x} \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
\[ y = -\frac{7}{8x} \]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, we swap the roles of \( x \) and \( y \):
\[ x = -\frac{7}{8y} \]

### Step 3: Solve for \( y \)
Now, solve this equation for \( y \):

1. Multiply both sides by \( 8y \):
   \[ 8y \cdot x = -7 \]

2. Divide both sides by \( 8x \) to isolate \( y \):
   \[ y = -\frac{7}{8x} \]

### Step 4: Replace \( y \) with \( f^{-1}(x) \)
The inverse function is:
\[ f^{-1}(x) = -\frac{7}{8x} \]

Thus, the inverse function of \( f(x) \) is:
\[
f^{-1}(x) = -\frac{7}{8x}
\]

### Verification
To verify if this is correct, we can check if applying \( f \) and \( f^{-1} \) to each other gives back the original value of \( x \):
- \( f(f^{-1}(x)) = -\frac{7}{8 \cdot \left( -\frac{7}{8x} \right)} = x \)
- \( f^{-1}(f(x)) = -\frac{7}{8 \cdot \left( -\frac{7}{8x} \right)} = x \)

The results confirm that the functions are inverses of each other.

Would you like more details or have any questions about this process?

Here are 5 related questions you might find helpful:
1. How do you determine if a function is invertible?
2. What are the conditions for a function to have an inverse?
3. Can the inverse function always be found for any given function?
4. How do you verify that two functions are inverses of each other?
5. How do inverse functions relate to symmetry about the line \( y = x \)?

**Tip:** To find the inverse of a function, remember to swap \( x \) and \( y \) and then solve for \( y \) to determine the new equation.

Find the inverse function of the function f(x) = -7/(8x)

This page explains how to find the inverse of the function f(x) = -7/(8x) with a step-by-step solution. Topics include inverse functions, algebraic manipulation, and verification using function composition. Suitable for students in Grades 10-12.

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