Find the inverse for each of the following functions.

a)  
k
(
x
)
=
15
x
−
5
 

k
−
1
(
x
)
=
  
  





b)  
f
(
x
)
=
9
√
x
x
+
7
 
   
f
−
1
(
x
)
=
   
  

### a) \( k(x) = 15x - 5 \)

To find the inverse function, follow these steps:

1. Start by replacing \( k(x) \) with \( y \):
   \[
   y = 15x - 5
   \]

2. Solve for \( x \) in terms of \( y \):
   \[
   y + 5 = 15x
   \]
   \[
   x = \frac{y + 5}{15}
   \]

3. Now, replace \( y \) with \( x \) to represent the inverse function:
   \[
   k^{-1}(x) = \frac{x + 5}{15}
   \]

### b) \( f(x) = 9\sqrt{x} + x + 7 \)

Finding the inverse of this function is more complex because it involves both a square root and a linear term. Let's attempt to solve it:

1. Replace \( f(x) \) with \( y \):
   \[
   y = 9\sqrt{x} + x + 7
   \]

2. Isolate the square root term, which is \( 9\sqrt{x} \), but in this case, it will be quite challenging due to the additional \( x \) and constant term. Unfortunately, this particular function's inverse cannot be expressed in a simple closed form due to its complexity.

Therefore, the inverse function of \( f(x) = 9\sqrt{x} + x + 7 \) cannot be solved algebraically. Numerical or approximation methods may be required for specific values of \( x \).

---

Would you like more details or have any questions on either function?

### Relative Questions:
1. How can you verify if the inverse function you found is correct?
2. What methods are typically used to find the inverse of a function involving both a radical and a polynomial term?
3. Can all functions have inverses? Why or why not?
4. How can you determine if a function is invertible by looking at its graph?
5. What are some numerical methods used for approximating inverses when algebraic methods fail?

#### Tip:
To check if a function has an inverse, it must be **one-to-one** and pass the **horizontal line test**.

Find the inverse for each of the following functions.
a) k(x) = 15x - 5
k−1(x) =

b) f(x) = 9√x + x + 7
f−1(x) =

Discover how to find the inverse of linear and complex functions. This detailed solution walks through the inverse of k(x) = 15x - 5, and why f(x) = 9√x + x + 7 cannot be easily inverted algebraically. Ideal for high school students learning algebra and inverse functions.

Math Problem Statement

a)
k ( x )

k − 1 ( x )

b)
f ( x )

f − 1 ( x )

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