To determine whether an inverse function exists for the given function $f(x) = x^2 - 16x + 64$, we need to analyze whether the function is one-to-one (injective). A function must be one-to-one to have an inverse, which means that for each output value, there is exactly one corresponding input value.

Step 1: Analyze the Function

The function $f(x) = x^2 - 16x + 64$ is a quadratic function. To determine if it is one-to-one, we can look at its graph or check if it passes the Horizontal Line Test: a function passes this test if any horizontal line intersects the graph at most once.

Step 2: Rewrite the Function in Vertex Form

We can rewrite the function in its vertex form to better understand its shape:

$f(x) = x^2 - 16x + 64$

Complete the square:

1. Take the coefficient of $x$, which is $-16$, halve it to get $-8$, and then square it to get 64.
2. Rewrite the function:

$f(x) = (x - 8)^2$

This shows that the function is a perfect square with its vertex at $(8, 0)$.

Step 3: Determine the Shape of the Graph

The function $(x - 8)^2$ is a parabola that opens upwards, which means it is not one-to-one over its entire domain because it fails the Horizontal Line Test.

Step 4: Restrict the Domain (if needed)

To have an inverse, the function needs to be restricted to a domain where it is either strictly increasing or decreasing. For $(x - 8)^2$:

• For $x \geq 8$, the function is strictly increasing.
• For $x \leq 8$, the function is strictly decreasing.

Thus, restricting the function to either $x \geq 8$ or $x \leq 8$ would allow the inverse to exist within those intervals.

Conclusion

The function $f(x) = x^2 - 16x + 64$ does not have an inverse over its entire domain but can have an inverse if its domain is restricted to $x \geq 8$ or $x \leq 8$.

Would you like more details on finding the inverse function within these restricted domains, or do you have any other questions?

Related Questions

1. How do you find the inverse of a quadratic function when the domain is restricted?
2. What are the steps to complete the square for any quadratic function?
3. Why does the Horizontal Line Test determine if a function has an inverse?
4. How do you determine the range of a quadratic function?
5. What is the significance of the vertex in determining the inverse of a quadratic function?

Tip

Always check the function’s one-to-one nature by using the Horizontal Line Test to determine if an inverse exists without domain restrictions.