Question

homework assignment 1.7: inverse functions
score: 4/15 answered: 4/15
question 5
(a) find the inverse function of \\( f(x) = 8x - 6 \\).
\\( f^{-1}(x) = \\)
(b) the graphs of \\( f \\) and \\( f^{-1} \\) are symmetric with respect to the line defined by \\( y = \\)

Explanation:

Part (a)

Step 1: Replace \( f(x) \) with \( y \)

We start with the function \( f(x) = 8x - 6 \). Replace \( f(x) \) with \( y \), so we have \( y = 8x - 6 \).

Step 2: Swap \( x \) and \( y \)

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = 8y - 6 \).

Step 3: Solve for \( y \)

First, add 6 to both sides of the equation: \( x + 6 = 8y \). Then, divide both sides by 8: \( y=\frac{x + 6}{8}\). We can also write this as \( y=\frac{1}{8}x+\frac{6}{8}=\frac{1}{8}x+\frac{3}{4}\).

Step 4: Replace \( y \) with \( f^{-1}(x) \)

Now, replace \( y \) with \( f^{-1}(x) \) to get the inverse function. So, \( f^{-1}(x)=\frac{x + 6}{8}\) (or \( \frac{1}{8}x+\frac{3}{4}\)).

Part (b)

The graphs of a function and its inverse are symmetric with respect to the line \( y = x \). This is a fundamental property of inverse functions. If we have a point \( (a,b) \) on the graph of \( f(x) \), then the point \( (b,a) \) will be on the graph of \( f^{-1}(x) \), and these two points are symmetric with respect to the line \( y=x \).

Answer:

(a) \( f^{-1}(x)=\boldsymbol{\frac{x + 6}{8}}\) (or \( \frac{1}{8}x+\frac{3}{4}\))
(b) \( y = \boldsymbol{x}\)