g(x)=5x-1, gh(x)=35x+19, find an expression for g-1 (x), find h(x)

Let's solve this step-by-step.

1. **Find the inverse of \( g(x) \)**:
   The given function is \( g(x) = 5x - 1 \).

   To find the inverse, \( g^{-1}(x) \):
   - Replace \( g(x) \) with \( y \): \( y = 5x - 1 \).
   - Swap \( x \) and \( y \): \( x = 5y - 1 \).
   - Solve for \( y \):

     \[
     x = 5y - 1
     \]

     \[
     x + 1 = 5y
     \]

     \[
     y = \frac{x + 1}{5}
     \]

   Therefore, the inverse function is \( g^{-1}(x) = \frac{x + 1}{5} \).

2. **Find \( h(x) \)**:
   Given \( gh(x) = g(h(x)) = 35x + 19 \) and \( g(x) = 5x - 1 \), we need to find \( h(x) \).

   Let's denote \( h(x) = y \). Then \( g(h(x)) = g(y) \):

   \[
   g(y) = 5y - 1
   \]

   According to the problem, \( g(y) = 35x + 19 \):

   \[
   5y - 1 = 35x + 19
   \]

   Solving for \( y \):

   \[
   5y = 35x + 20
   \]

   \[
   y = 7x + 4
   \]

   Therefore, \( h(x) = 7x + 4 \).

**Summary**:
- The inverse function \( g^{-1}(x) = \frac{x + 1}{5} \).
- The function \( h(x) = 7x + 4 \).

Do you want details or have any questions? Here are some additional questions for further practice:

1. If \( h(x) = 3x - 2 \), what is \( gh(x) \)?
2. Find the inverse of the function \( h(x) = 2x + 3 \).
3. Verify that \( g(g^{-1}(x)) = x \) for \( g(x) = 5x - 1 \).
4. If \( f(x) = x^2 \) and \( g(x) = 3x + 2 \), find the composition \( f(g(x)) \).
5. Determine the value of \( x \) for which \( g(x) = 14 \) if \( g(x) = 5x - 1 \).

**Tip**: Always check your solution by substituting it back into the original equations to ensure it satisfies all given conditions.

Learn how to find the inverse function of g(x) = 5x - 1 and determine h(x) for the given function gh(x) = 35x + 19. This problem involves functions and inverse functions, suitable for advanced high school students.

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