Problem 31 - Algebra and Trigonometry

Question

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ \begin{array}{l}{f(x)=x^{2}-4, \quad x \geq 0} \\ {g(x)=\sqrt{x+4}, \quad x \geq-4}\end{array} $$

Step-by-Step Solution

Answer

The functions \(f(x) = x^2 - 4\) and \(g(x) = \sqrt{x+4}\) are inverses of each other.

Step 1: Understand the Inverse Function Property

Two functions \(f\) and \(g\) are inverses of each other if and only if \(f(g(x)) = x\) and \(g(f(x)) = x\) for every \(x\) in the domain of the corresponding function.

Step 2: Verify \(f(g(x)) = x\)

Start by substituting \(g(x)\) into \(f(x)\). This means finding \(f(\sqrt{x+4})\). Substitute \(g(x)\) into \(f(x)\) as follows: \[f(g(x)) = f(\sqrt{x+4}) = (\sqrt{x+4})^2 - 4\]Simplify the expression:\[(\sqrt{x+4})^2 = x+4\]So we have:\[x+4-4=x\]Thus, \(f(g(x)) = x\).

Step 3: Verify \(g(f(x)) = x\)

Substitute \(f(x)\) into \(g(x)\). This means finding \(g(x^2-4)\). Substitute \(f(x)\) into \(g(x)\) as follows:\[g(f(x)) = g(x^2 - 4) = \sqrt{(x^2-4)+4}\]Simplify the expression:\[\sqrt{x^2} = x\text{ (since } x \geq 0\text{)}\]Thus, \(g(f(x)) = x\).

Step 4: Conclusion

Since we have shown that both \(f(g(x)) = x\) and \(g(f(x)) = x\), the functions \(f(x) = x^2 - 4\) and \(g(x) = \sqrt{x+4}\) are indeed inverses of each other.

Key Concepts

Composition of Functions

The composition of functions involves making one function the input of another. This process is often denoted by \(f(g(x))\) or \(g(f(x))\). To determine whether two functions are inverses of each other, you need to check if the composition of these functions returns the original input, \(x\). Essentially, this means: \(\begin{align*}&f(g(x)) = x\ &g(f(x)) = x\end{align*}\)When dealing with inverse functions, if these two conditions hold true for all \(x\) in the domains of their respective compositions, then the functions are truly inverses. This concept not only checks compatibility but also helps explore the underlying relationships between two functions.

Domain and Range

Domain and range are fundamental concepts describing the inputs and outputs of a function. The domain of a function consists of all possible input values, while the range includes all potential outputs.

• For \(f(x) = x^2 - 4\), the domain is \(x \geq 0\), meaning it accepts non-negative real numbers.
• For \(g(x) = \sqrt{x+4}\), the domain is \(x \geq -4\), as the expression under the square root must be non-negative.

The correct choice of domain and range ensures the functions are working within valid values. Additionally, when proving if two functions are inverses, these need to be considered to confirm their complete compatibility for the reversibility essential in inverse functions.

Square Root Function

A square root function is represented as \(g(x) = \sqrt{x}\). It is only defined for non-negative values to avoid imaginary numbers. The principal characteristic of the square root function is its ability to "undo" a squaring operation within its defined domain.

• For example, in the exercise, \(g(x) = \sqrt{x+4}\) takes the output from a quadratic function and transforms it back to the original input.
• Here, the modification involves adding a constant, meaning the domain starts at \(x \geq -4\) to ensure only non-negative values go into the square root.

Understanding square root functions is crucial as they appear often when discussing the principles of inverse functions. They demonstrate how a function can reverse the effects of another function when tailored correctly within their domains.

Quadratic Function

A quadratic function like \(f(x) = x^2 - 4\) is a polynomial of degree two. It is characterized by a parabolic graph that opens either upwards or downwards depending on the leading coefficient.

• Here, the function is modified by subtracting a constant, indicating a downward shift of the graph.
• Since we specifically consider \(x \geq 0\), this ensures the function always returns non-negative values.

The behavior of quadratic functions features prominently in finding inverse functions. When properly constrained (for example, by limiting the domain to non-negative values), the quadratic successfully relates to square root functions. Thus, together, they form inverse relations, as shown in the given exercise.