(a) Find the inverse of $f(x) = 8x - 2$.

To find the inverse, follow these steps:

1. Replace $f(x)$ with $y$: $y = 8x - 2$

2. Swap $x$ and $y$ to find the inverse function: $x = 8y - 2$

3. Solve for $y$ (the inverse function): $x + 2 = 8y$ $y = \frac{x + 2}{8}$ So, the inverse function is: $f^{-1}(x) = \frac{x + 2}{8}$

4. Check the answer: To verify, we need to show that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

   • First, substitute $f^{-1}(x)$ into $f(x)$: $f\left( f^{-1}(x) \right) = f\left( \frac{x + 2}{8} \right) = 8 \cdot \frac{x + 2}{8} - 2 = x + 2 - 2 = x$
   • Now, substitute $f(x)$ into $f^{-1}(x)$: $f^{-1}(f(x)) = f^{-1}(8x - 2) = \frac{(8x - 2) + 2}{8} = \frac{8x}{8} = x$ Since both hold true, the inverse is correct.

Thus, the inverse function is: $f^{-1}(x) = \frac{x + 2}{8}$

(b) Find the domain and range of $f(x)$ and $f^{-1}(x)$.

• The domain of $f(x)$ is all real numbers $\mathbb{R}$ because $8x - 2$ is a linear function, which is defined for all real numbers. $\text{Domain of } f(x) = \mathbb{R}$

• The range of $f(x)$ is also all real numbers $\mathbb{R}$, since as $x$ takes any real value, $8x - 2$ can output any real number. $\text{Range of } f(x) = \mathbb{R}$

• The domain of $f^{-1}(x)$ is the range of $f(x)$, which is $\mathbb{R}$. $\text{Domain of } f^{-1}(x) = \mathbb{R}$

• The range of $f^{-1}(x)$ is the domain of $f(x)$, which is $\mathbb{R}$. $\text{Range of } f^{-1}(x) = \mathbb{R}$

(c) Graph $f(x)$, $f^{-1}(x)$, and $y = x$ on the same coordinate axes.

To sketch the graphs:

1. Graph of $f(x) = 8x - 2$: This is a straight line with a slope of 8 and a y-intercept at -2.

2. Graph of $f^{-1}(x) = \frac{x + 2}{8}$: This is a straight line with a slope of $\frac{1}{8}$ and a y-intercept at $\frac{2}{8} = 0.25$.

3. Graph of $y = x$: This is the line that passes through the origin with a slope of 1, serving as the reflection line for the function and its inverse.

Would you like me to generate a graph of these functions?

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Relative Questions:

1. How do you verify if a function is one-to-one?
2. What conditions are necessary for a function to have an inverse?
3. How do you determine the domain and range of composite functions?
4. Can a quadratic function have an inverse? Why or why not?
5. How do you interpret the graph of $y = x$ when comparing a function and its inverse?

Tip:

When finding an inverse, always check your result by composing the function and its inverse to ensure that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.