Problem 27 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Define the Inverse Function Property

Two functions \( f \) and \( g \) are inverses of each other if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the domains of the respective compositions.

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

Substitute \( g(x) = \sqrt{x + 4} \) into \( f(x) = x^2 - 4 \): \[ f(g(x)) = f(\sqrt{x + 4}) = (\sqrt{x + 4})^2 - 4 = x + 4 - 4 = x \]This confirms that \( f(g(x)) = x \) for the domain \( x \geq -4 \).

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

Substitute \( f(x) = x^2 - 4 \) into \( g(x) = \sqrt{x + 4} \): \[ g(f(x)) = g(x^2 - 4) = \sqrt{(x^2 - 4) + 4} = \sqrt{x^2} = x \]This confirms that \( g(f(x)) = x \) for the domain \( x \geq 0 \).

Step 4: Conclusion

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true for their respective domains, \( f(x) = x^2 - 4 \) and \( g(x) = \sqrt{x+4} \) are indeed inverses of each other.

Key Concepts

Inverse Function Property

When exploring the concept of inverse functions, the Inverse Function Property is a fundamental tool. It tells us that two functions, say \( f \) and \( g \), are inverses if they "undo" each other. This means if you plug \( g(x) \) into \( f(x) \) and vice versa, you should get back to your original input \( x \).

In mathematical terms, this is described by the equations \( f(g(x)) = x \) and \( g(f(x)) = x \). If both of these conditions are satisfied for all \( x \) in the appropriate domains, then \( f \) and \( g \) can be confirmed as inverse functions.

• Substitute \( g(x) \) into \( f(x) \) and simplify. If you end up with \( x \), you've completed half the verification.
• Next, substitute \( f(x) \) into \( g(x) \) and simplify. Arriving at \( x \) here, too, confirms they are inverses.

compositions of functions

The composition of two functions involves taking the output of one function and using it as the input for another. This is expressed as \( f(g(x)) \) or \( g(f(x)) \).

For example, if you have the function \( f(x) = x^2 - 4 \) and \( g(x) = \sqrt{x + 4} \), the composition \( f(g(x)) \) means inserting \( g(x) \) in place of \( x \) in \( f(x) \). Similarly, \( g(f(x)) \) requires inserting \( f(x) \) into \( g(x) \).

• Functions \( f(x) \) and \( g(x) \) "undo" each other's operations when composed, confirming their inverse nature.
• These compositions need to work within the domains of the functions. This ensures the compositions make sense mathematically.

domain and range of functions

Understanding the domain and range is crucial when dealing with functions and their inverses. The domain of a function is the complete set of possible values of \( x \) which will make the function work, while the range is the set of possible outputs.

For \( f(x) = x^2 - 4 \), the specified domain is \( x \geq 0 \), meaning \( f \) can only handle non-negative inputs in this context. Meanwhile, for \( g(x) = \sqrt{x+4} \), the domain is \( x \geq -4 \), accommodating the result of \( x^2-4 \).

• Domains and ranges must align properly for \( f \) and \( g \) to suffice as true inverses.
• Without correct domain considerations, the function compositions \( f(g(x)) \) and \( g(f(x)) \) might not always return meaningful results.