Problem 34 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=(x-4)^{2}, \quad x \geq 4 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{x} + 4 \).

Step 1: Understand the Function

The function given is \( f(x) = (x-4)^2 \) with the domain \( x \geq 4 \). This means it is defined for all \( x \) greater than or equal to 4, and outputs squared values.

Step 2: Set Up the Equation for Inversion

To find the inverse, swap \( x \) and \( y \) in the function, then solve for \( y \). Start with \( y = (x-4)^2 \), then write it as \( x = (y-4)^2 \).

Step 3: Solve for the Inverse

To solve for \( y \), take the square root of both sides: \[ \sqrt{x} = y - 4 \]. Since the original domain is \( x \geq 4 \), we consider the positive square root. So, we have: \[ y = \sqrt{x} + 4 \].

Step 4: Rewrite the Inverse Function

Thus, the inverse function is \( f^{-1}(x) = \sqrt{x} + 4 \) with the domain \( x \geq 0 \).

Step 5: Graph the Function and its Inverse

Graph the original function \( f(x) = (x-4)^2 \) and its inverse \( f^{-1}(x) = \sqrt{x} + 4 \). The original graph is a parabola opening upwards starting at (4,0), and the inverse is a square root function shifted right by 4.

Key Concepts

Function Graphing

When graphing functions and their inverses, it's important to visualize how they relate to each other. A function and its inverse are reflections of each other over the line \(y = x\). This means if you have a point \((a, b)\) on the original function, the inverse function will have the point \((b, a)\).

• The original function \(f(x) = (x-4)^2\) graphs as a parabola that opens upwards.
• This parabola starts at point (4,0), due to the domain restriction \(x \geq 4\).
• The inverse \(f^{-1}(x) = \sqrt{x} + 4\) shifts the familiar square root curve to the right by 4 units, starting at (0,4).

While graphing, keep in mind the domain and range of the function and its inverse to properly position them on the coordinate plane. The inverse essentially undoes the action of the original function by reflecting it, providing fascinating symmetry.

Function Domains

The domain of a function is the set of all possible input values (usually \(x\)-values) to the function. Understanding domains is crucial since they determine what part of the graph actually exists.

• For the function \(f(x) = (x-4)^2\), the domain is \(x \geq 4\).
• This means the function is only defined for values of \(x\) that are 4 or greater, and you will only draw the graph starting from \(x = 4\).
• The inverse function \(f^{-1}(x) = \sqrt{x} + 4\) has a domain of \(x \geq 0\), meaning it is defined for all non-negative \(x\).

Domain considerations are key because they affect how functions are graphed and interpreted. Without proper domain analysis, one might mistakenly graph or solve beyond the intended limits of the function.

Square Root Functions

Square root functions have unique properties and graphing characteristics. The general form is \(f(x) = \sqrt{x}\), but transformations can adjust its shape and position.

• In our example, \(f^{-1}(x) = \sqrt{x} + 4\) involves a horizontal shift.
• This function shifts the basic square root graph right by 4 units, starting at the point (0,4) due to the domain \(x \geq 0\).
• Square root functions are typically half of a parabola, only taking on non-negative \(y\)-values.

These functions always start from the origin unless shifted, and they increase gradually. Understanding their graph helps in visualizing how inputs are transformed and in interpreting the function's behavior over its domain.