Problem 35 - Algebra for College Students

Question

Determine whether \(f\) and \(g\) are inverse functions. $$ f(x)=\sqrt{x+1} \text { and } g(x)=x^{2}-1 \quad \text { for } x \geq 0 $$

Step-by-Step Solution

Answer

Yes, \( f \) and \( g \) are inverse functions.

Step 1: Understand Inverse Functions

Two functions \( f \) and \( g \) are inverses of each other if the composition of \( f(g(x)) \) equals \( x \) and \( g(f(x)) \) also equals \( x \) for all \( x \) in the domains of \( g \) and \( f \), respectively.

Step 2: Check \( f(g(x)) \)

Calculate \( f(g(x)) \): \[ f(g(x)) = f(x^2 - 1) = \sqrt{(x^2 - 1) + 1} = \sqrt{x^2} = x \] For \( x \geq 0 \), \( \sqrt{x^2} = x \), so \( f(g(x)) = x \).

Step 3: Check \( g(f(x)) \)

Calculate \( g(f(x)) \): \[ g(f(x)) = g(\sqrt{x+1}) = (\sqrt{x+1})^2 - 1 = (x+1) - 1 = x \]For \( x \geq 0 \), this expression simplifies to \( x \), so \( g(f(x)) = x \).

Step 4: Draw Conclusion

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \) for \( x \geq 0 \), functions \( f \) and \( g \) are inverses of each other.

Key Concepts

Composition of Functions

The concept of composition of functions is essential for determining inverse functions. A composition involves taking one function and applying it to the result of another function. This is denoted as \( f(g(x)) \) or \( g(f(x)) \), where we substitute the output of the second function into the first. In this exercise, the goal is to check if the composition results in the identity function, meaning the original input \( x \) is obtained back.
Understanding how to properly substitute and simplify these compositions is crucial.

• Start with substituting the inner function into the outer function.
• Simplify the composition to check if it results in \( x \).
• If both \( f(g(x)) \) and \( g(f(x)) \) equal \( x \), functions \( f \) and \( g \) are inverses.

This approach confirms that the operations of each function effectively "undo" each other, a characteristic of inverse functions.

Square Root Function

The square root function is a fundamental math concept, represented as \( f(x) = \sqrt{x} \). Here, \( f(x) = \sqrt{x + 1} \) is used, slightly more complex than the basic form. The square root function typically undoes squaring by taking positive square roots. It’s crucial to pay attention to the domain when dealing with square roots because we only consider non-negative outputs in standard real numbers.
In this specific problem, the domain ensures that we handle the function correctly:

• The function \( f(x) = \sqrt{x + 1} \) is applied to the result of another function \( g(x) \).
• Make sure to simplify the expression under the square root first, then take its square root.

Conceptually, when tests show \( f(g(x)) = x \) and it holds for the domain, it aligns with the properties of the square root function as an inverse in this context.

Quadratic Function

Quadratic functions are expressed in their general form \( g(x) = ax^2 + bx + c \). In this problem, it is simplified as \( g(x) = x^2 - 1 \). Quadratic functions have a fundamental role in inverse function determination since they involve squaring, and many functions concerned with inverses will involve both squaring and square roots.
A key property of quadratic functions is their parabolic shape, which can reverse operations like addition and subtraction.

• The quadratic function \( g(x) = x^2 - 1 \) effects an inverse test to see if when composed with \( f \), it returns \( x \).
• Simplification of \( g(f(x)) \) involves replacing \( x \) with \( \sqrt{x + 1} \), followed by squaring and subtracting 1.

If \( g(f(x)) = x \) after simplification, it confirms that the quadratic function effectively reverses the operation of the square root function in this domain, contributing to identifying it as an inverse.