(a) Find the inverse of $f(x) = 16x - 4$

To find the inverse, follow these steps:

1. Start by writing $y = f(x) = 16x - 4$.
2. Swap $x$ and $y$ to reflect the inverse function:
   $x = 16y - 4$.
3. Solve for $y$: $x + 4 = 16y$ $y = \frac{x + 4}{16}$ So, the inverse function is $f^{-1}(x) = \frac{x + 4}{16}$.

Check the inverse:

To verify, we check if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• First, $f(f^{-1}(x))$: $f\left(f^{-1}(x)\right) = f\left(\frac{x + 4}{16}\right) = 16\left(\frac{x + 4}{16}\right) - 4 = x + 4 - 4 = x$
• Second, $f^{-1}(f(x))$: $f^{-1}(f(x)) = f^{-1}(16x - 4) = \frac{(16x - 4) + 4}{16} = \frac{16x}{16} = x$

Both compositions return $x$, so the inverse is correct.

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

• Domain of $f(x) = 16x - 4$: Since $f(x)$ is a linear function, the domain is all real numbers:
  $\text{Domain of } f = (-\infty, \infty)$
• Range of $f(x)$: As a linear function, $f(x)$ can take any real value, so the range is:
  $\text{Range of } f = (-\infty, \infty)$
• Domain of $f^{-1}(x) = \frac{x + 4}{16}$: The inverse is also linear, so the domain is all real numbers:
  $\text{Domain of } f^{-1} = (-\infty, \infty)$
• Range of $f^{-1}(x)$: Since the inverse function is linear, the range is also all real numbers:
  $\text{Range of } f^{-1} = (-\infty, \infty)$

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

To graph these three functions:

1. $f(x) = 16x - 4$: This is a straight line with slope 16 and y-intercept -4.
2. $f^{-1}(x) = \frac{x + 4}{16}$: This is a straight line with slope $\frac{1}{16}$ and y-intercept $\frac{1}{4}$.
3. $y = x$: This is a diagonal line passing through the origin with a slope of 1.

The graph of $f(x)$ and $f^{-1}(x)$ should be symmetric about the line $y = x$.

Let me know if you'd like me to generate a graph.

Follow-up Questions:

1. Can you explain why the graph of a function and its inverse are symmetric about the line $y = x$?
2. How would you find the inverse of a non-linear function?
3. Why is it important to check both compositions when verifying the inverse function?
4. Could a function that is not one-to-one have an inverse?
5. What is the geometric meaning of the domain and range in a function’s graph?

Tip:

Always remember that a function must be one-to-one (pass the horizontal line test) to have an inverse that is also a function.