Problem 26 - Algebra for College Students

Question

Verify that the two given functions are inverses of each other. \(f(x)=x^{2}-4\) for \(x \geq 0\) and $$ g(x)=\sqrt{x+4} \text { for } x \geq-4 $$

Step-by-Step Solution

Answer

The functions are inverses of each other.

Step 1: Verify if Composition of Functions Equals Identity for f(g(x))

First, we need to check if \( f(g(x)) = x \) for all \( x \) in the domain of \( g \).\\[ f(g(x)) = f(\sqrt{x+4}) \]\Substitute \( g(x) \) into \( f(x) \): \\[ f(\sqrt{x+4}) = (\sqrt{x+4})^2 - 4 \]\\[= x + 4 - 4 = x \]\Since \( f(g(x)) = x \), this condition is satisfied.

Step 2: Verify if Composition of Functions Equals Identity for g(f(x))

Now, we check if \( g(f(x)) = x \) for all \( x \) in the domain of \( f \).\\[ g(f(x)) = g(x^2 - 4) \]\Substitute \( f(x) \) into \( g(x) \):\\[ g(x^2 - 4) = \sqrt{(x^2 - 4) + 4} \]\\[ = \sqrt{x^2} = x \] \ Since \( g(f(x)) = x \) for all \( x \geq 0 \), this condition is satisfied.

Key Concepts

Function Composition

Function composition involves applying one function to the results of another. It allows us to build complex operations from simpler ones by feeding the output of one function as the input to another. For example, saying "compose functions \( f \) and \( g \)" means to create a new function, \( f(g(x)) \), where \( g \) is applied first and then \( f \) is applied to the result of \( g \).
Think of it like a process chain, where each step depends on the outcome of the previous step.
For the given functions, we verified \( f(g(x)) = x \) and \( g(f(x)) = x \). This indicates that each function undoes the operations of the other, which is a characteristic of inverse functions.

• When \( f(g(x)) = x \), \( g \) is said to "undo" \( f \).
• Similarly, when \( g(f(x)) = x \), \( f \) "undoes" \( g \).

This mutual cancellation shows that the functions truly are inverses of one another. Function composition not only checks for this relationship but also reveals deeper insights about function interactions.

Domain and Range

The domain of a function is the set of input values for which the function is defined. The range, on the other hand, is the set of possible outputs the function can produce.
When dealing with inverse functions, the domain and range play vital roles. This is because the domain of \( f \) becomes the range of \( g \), and the range of \( f \) becomes the domain of \( g \).
For the function \( f(x) = x^2 - 4 \), the domain is \( x \geq 0 \) because we are only considering the non-negative side of the square function. Its range would be all possible outputs, \( y \geq -4 \), since the smallest value is when \( x = 0 \).

• Domain of \( f \): \( x \geq 0 \)
• Range of \( f \): \( y \geq -4 \)

For \( g(x) = \sqrt{x + 4} \), the domain is \( x \geq -4 \), since these inputs result in non-negative numbers under the square root operation. The range would be \( y \geq 0 \), as the square root function always yields non-negative results.

• Domain of \( g \): \( x \geq -4 \)
• Range of \( g \): \( y \geq 0 \)

This reciprocal relationship between the domain and range of inverse functions confirms their status as inverses, making domains and ranges crucial for understanding function behaviors.

Square Root Function

The square root function, \( g(x) = \sqrt{x} \), maps a non-negative input to its square root. Mathematically, it's defined for \( x \geq 0 \). The function deals exclusively with the principal (non-negative) square root.
Understanding the behavior of square root functions is essential in many areas of mathematics, especially when determining inverse functions.
In our exercise, \( g(x) = \sqrt{x + 4} \). This adjusts the standard square root function to shift horizontally. The function is defined for all \( x \geq -4 \), ensuring every input under the square root produces a real number.

• Shifting to \( x + 4 \) adjusts the start point from 0 to -4.
• Output starts at 0, increasing as \( x \) increases.

Since square roots are non-negative, the range begins at 0 and increases for larger values. This complements the function \( f(x) = x^2 - 4 \), whose range starts at -4, exactly matching \( g \)'s domain.
By visualizing and understanding these operations, it's clearer how square roots fit into the larger puzzle of inverse functions.