The function f left parenthesis x right parenthesis equals 6 x plus 5 is​ one-to-one.
a. Find an equation for f Superscript negative 1​, the inverse function.
b. Verify that your equation is correct by showing that f left parenthesis f Superscript negative 1 Baseline left parenthesis x right parenthesis right parenthesisequalsx and f Superscript negative 1 Baseline left parenthesis f left parenthesis x right parenthesis right parenthesisequalsx.

Let's work through the problem step by step.

### a. Find the equation for the inverse function \( f^{-1}(x) \)

We are given that \( f(x) = 6x + 5 \). To find the inverse, follow these steps:

1. Replace \( f(x) \) with \( y \): 
   \[
   y = 6x + 5
   \]
2. Swap \( x \) and \( y \) to solve for the inverse:
   \[
   x = 6y + 5
   \]
3. Solve for \( y \) (this will be \( f^{-1}(x) \)):
   \[
   x - 5 = 6y
   \]
   \[
   y = \frac{x - 5}{6}
   \]
So, the inverse function is:
\[
f^{-1}(x) = \frac{x - 5}{6}
\]

### b. Verify the inverse function

We need to verify that:
1. \( f(f^{-1}(x)) = x \)
2. \( f^{-1}(f(x)) = x \)

#### 1. Verify \( f(f^{-1}(x)) = x \)
Substitute \( f^{-1}(x) = \frac{x - 5}{6} \) into \( f(x) \):
\[
f(f^{-1}(x)) = 6\left( \frac{x - 5}{6} \right) + 5
\]
Simplify:
\[
= (x - 5) + 5 = x
\]
So, \( f(f^{-1}(x)) = x \).

#### 2. Verify \( f^{-1}(f(x)) = x \)
Substitute \( f(x) = 6x + 5 \) into \( f^{-1}(x) \):
\[
f^{-1}(f(x)) = \frac{(6x + 5) - 5}{6}
\]
Simplify:
\[
= \frac{6x}{6} = x
\]
So, \( f^{-1}(f(x)) = x \).

Both conditions are satisfied, confirming that the inverse function \( f^{-1}(x) = \frac{x - 5}{6} \) is correct.

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the graphical interpretation of the inverse of a function?
2. How do you determine if a function is one-to-one without graphing it?
3. How does the horizontal line test relate to one-to-one functions?
4. What are the conditions for a function to have an inverse?
5. How do we handle finding the inverse of a quadratic function?

**Tip:** To quickly check if a function is one-to-one, see if it passes the horizontal line test: no horizontal line should intersect the graph more than once.

The function f(x) = 6x + 5 is one-to-one. a. Find an equation for f⁻¹, the inverse function. b. Verify that your equation is correct by showing that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Learn how to find the inverse of the linear function f(x) = 6x + 5 and verify it step-by-step. This problem involves key algebraic concepts such as inverse functions and verification through substitution. Suitable for high school students in Grades 10-12.

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