Find the inverse function of the function f, of, x, equals, start fraction, 7, x, divided by, 5, end fraction*f*(*x*)=57*x*​.

To find the inverse function of \( f(x) = \frac{7x}{5} \), let's follow these steps:

1. **Replace \( f(x) \) with \( y \):**
   \[
   y = \frac{7x}{5}
   \]

2. **Swap \( x \) and \( y \):**
   \[
   x = \frac{7y}{5}
   \]

3. **Solve for \( y \):**
   To solve for \( y \), multiply both sides by \( \frac{5}{7} \):
   \[
   y = \frac{5}{7}x
   \]

4. **Write the inverse function:**
   The inverse function \( f^{-1}(x) \) is:
   \[
   f^{-1}(x) = \frac{5}{7}x
   \]

So, the inverse function of \( f(x) = \frac{7x}{5} \) is:

\[
f^{-1}(x) = \frac{5}{7}x
\]

Would you like any further details or have any questions? Here are some related questions to explore:

1. How do you verify that \( f \) and \( f^{-1} \) are indeed inverses?
2. What is the inverse of a function if it involves both multiplication and addition, like \( f(x) = 2x + 3 \)?
3. Can every function have an inverse? Why or why not?
4. How do you find the inverse of a quadratic function?
5. What does it mean geometrically for two functions to be inverses?

**Tip:** The graph of a function and its inverse are reflections of each other over the line \( y = x \).

Find the inverse function of the function f(x) = 7x/5.

Discover how to find the inverse function of f(x) = 7x/5 in a clear step-by-step solution. This problem involves key algebraic concepts such as functions and inverse functions. Ideal for students in Grades 9-10.

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