Problem 80 - Algebra 2

Question

Find the inverse of each function. Is the inverse a function? \(f(x)=\sqrt{x+3}-4\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x)=(x+4)^2 - 3\). Yes, it is a function as it passes the vertical line test.

Step 1: Swap the variables

Replace \(f(x)\) with \(y\), then swap \(x\) and \(y\). We get \(x=\sqrt{y+3}-4\)

Step 2: Solve for y

Rearrange the equation to solve for \(y\). That requires to isolate \(y\) on one side of the equation. To do so, the first step is to cancel out the -4 by adding 4 to both sides: \(x+4=\sqrt{y+3}\). Then square both sides of the equation to remove the square root: \((x+4)^2=y+3\). Finally subtract 3 from each side to isolate \(y\): \(y=(x+4)^2 - 3\)

Step 3: Check if the inverse is a function

To determine whether the inverse is a function, apply the vertical line test.The vertical line test states that if we can draw any vertical line that intersects a graph more than once, then the graph doesn't represent a function. Here, the graph of \(y=(x+4)^2-3\) is a parabola opening upwards, meaning any vertical line will intersect it at most once. Hence, the inverse function is a function.

Key Concepts

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. In simpler terms, it's like a chain reaction. If you have two functions, say \( f(x) \) and \( g(x) \), the composition is written as \( (f \circ g)(x) = f(g(x)) \). This means you first apply \( g \) to \( x \) and then apply \( f \) to the outcome.

• First step: Apply \( g(x) \) to an input \( x \).
• Second step: Take the output from \( g(x) \) and apply \( f \) to it.

Function composition is important in the study of inverses because an inverse function "undoes" the work of a function. When you compose a function and its inverse, \( (f \circ f^{-1})(x) \) or \( (f^{-1} \circ f)(x) \), the result will be the original input \( x \), assuming the inverse exists. Understanding this concept fully is essential in identifying and verifying inverse functions.

Vertical Line Test

The vertical line test is a simple yet powerful tool used to determine whether a graph represents a function. According to this test, if you can draw any vertical line that crosses your graph more than once, then the graph does not represent a function. A useful trick is to imagine a vertical ruler moving across the graph:

• If at any point the ruler touches the graph more than once, then it is not a function.
• If it only touches once all the time, then it is a function.

This test applies to any graph, including inverses. For example, when we drew the graph of the inverse function \( y=(x+4)^2-3 \) from our exercise, it was a parabola. Because no vertical line can intersect the curve more than once, it passes the vertical line test and confirms that the inverse is indeed a function.

Parabolas

Parabolas are U-shaped graphs that can open upwards or downwards. They are associated with quadratic equations of the form \( y = ax^2 + bx + c \). Here are some important properties of parabolas:

• Vertex: The point where the parabola changes direction. It is the minimum point when the parabola opens upwards and the maximum point when opening downwards.
• Axis of Symmetry: A vertical line that passes through the vertex, splitting the parabola into two identical halves.

In our solution, the inverse function \( y=(x+4)^2-3 \) describes a parabola that is shifted from the origin. This is crucial because it opens upwards, which aligns with the rule that ensures it passes the vertical line test, confirming the inverse is a function.