Problem 33 - College Algebra

Question

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$f(x)=(x+3)^{2}, x \geq-3$$

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = \sqrt{x} - 3 \) with domain \( x \geq 0 \).

Step 1: Understand the Problem

We have a function defined as \( f(x) = (x+3)^2 \) with the domain \( x \geq -3 \). We need to find its inverse and graph both the function and the inverse.

Step 1: Write the Function as y

Start by expressing the function with \( y \):\[ y = (x + 3)^2 \] Our goal is to solve for \( x \) in terms of \( y \).

Step 2: Solve for x in terms of y

To find the inverse, take the square root of both sides:\[ \sqrt{y} = x + 3 \] Since \( x \geq -3 \), we only take the positive root. Then solve for \( x \): \[ x = \sqrt{y} - 3 \]

Step 3: Swap Variables to Find Inverse

To express the inverse function, swap \( x \) and \( y \). This gives us \[ y = \sqrt{x} - 3 \]. Hence, the inverse function is \( f^{-1}(x) = \sqrt{x} - 3 \).

Step 4: Determine the Domain of Inverse

Identify the domain of the inverse function by looking at the range of the original function. Since \( f(x) = (x+3)^2 \) where \( x \geq -3 \), the range is \([0, \infty)\), so the domain of \( f^{-1}(x) \) is \( x \geq 0 \).

Step 6: Graph the Functions

Graph both the function \( f(x) = (x+3)^2 \) and its inverse \( f^{-1}(x) = \sqrt{x} - 3 \) over their appropriate domains. The graph of \( y = (x+3)^2 \) is a parabola opening upwards starting at \( x = -3 \), and \( y = \sqrt{x} - 3 \) is a square root function starting at \( x = 0 \). Both graphs reflect over the line \( y = x \).

Key Concepts

Function Graphing

Graphing functions is a visual way to understand how a function behaves across different values. For the function \( f(x) = (x+3)^2 \), its graph is a parabola. When graphing its inverse \( f^{-1}(x) = \sqrt{x} - 3 \), it's essential to keep in mind the restrictions on their domains and ranges.

• The original function's graph is a U-shaped curve, starting at the point \((-3, 0)\) and opening upwards.
• The inverse function's graph begins at the origin (0,0) but lowered by 3, following the form typical to square root functions.

These graphs help visualize how the original function transforms into its inverse, showing the reflection over the line \( y = x \). This reflection line is a great guide to understanding inverse relationships visually.

Domain and Range

The domain and range are key concepts in understanding a function's behavior. For \( f(x) = (x+3)^2 \), the domain is restricted to \(x \geq -3\) as indicated in the problem statement. This means it only takes values at or greater than -3. Conversely, the range of this function is all non-negative numbers, or \([0, \infty)\), because as you input numbers, the output \((x+3)^2\) can never be negative.

• The inverse function \( f^{-1}(x) = \sqrt{x} - 3 \) has a domain of \(x \geq 0\) since you cannot take the square root of a negative number.
• Its range is similar to the original function's domain, which is \([-3, \infty)\).

Understanding these sets of permissible inputs and outputs helps clarify how functions and their inverses interact.

Parabola

A parabola is a unique and important shape in mathematics. The function \( f(x) = (x+3)^2 \) is a classic example, where the graph forms this familiar U shape. Below are some key characteristics of parabolas:

• The vertex of the parabola \((x+3)^2\) occurs at the point \((-3, 0)\).
• It opens upward, meaning all of its values are positive or zero.
• Parabolas are symmetrical around their vertex, in this case, the vertical line \(x = -3\).

By analyzing the parabola graph, you can infer significant details about the function's minimum value and overall behavior. It's also useful to understand how this shape transfers to the inverse function graph.

Square Root Function

The square root function is the inverse of a quadratic function like \( f(x) = (x+3)^2 \). Its graph \( f^{-1}(x) = \sqrt{x} - 3 \) shows a rightward growth starting at \(x = 0\), depicting how outputs (y-values) change based on inputs (x-values). Here are some typical characteristics:

• It starts at \(y = -3\) when \(x = 0\).
• The curve rises slowly as \(x\) increases, illustrating the square root's nature of producing smaller outputs for equal increments of input.
• Just like all square root functions, \( \sqrt{x} \), it never produces negative outputs.

Understanding the square root graph aids in visualizing how quadratics get reversed through their inverse operations. It emphasizes gradual growth compared to the rapid increase of the original quadratic function.