Question

Jocelyn and Gerry determine that the inverse of the function \(f(x)=x^{2}-5, x \geq 0\) is \(f^{-1}(x)=\sqrt{x+5} .\) Does the graph verify that these functions are inverses of each other? Explain why.

Short answer

Yes, the graphs of the functions are inverses as they are symmetrical along the line \( y = x \).

Step-by-step solution

Understanding the Functions

The given functions are: - The original function: \( f(x) = x^2 - 5 \) with the domain \( x \geq 0 \) - The proposed inverse function: \( f^{-1}(x) = \sqrt{x + 5} \) The question asks to verify if these functions are indeed inverses of each other by analyzing their graphs.

Finding Composition of Functions

To verify if the functions are inverses, check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).Calculate \( f(f^{-1}(x)) \): \[ f(f^{-1}(x)) = f(\sqrt{x+5}) \] Substitute \( \sqrt{x+5} \) into the function \( f(x) \): \[ f(\sqrt{x+5}) = (\sqrt{x+5})^2 - 5 \] Simplify: \[ (\sqrt{x+5})^2 - 5 = x + 5 - 5 = x \] Therefore, \( f(f^{-1}(x)) = x \).

Finding the Other Composition

Now, calculate \( f^{-1}(f(x)) \): \[ f^{-1}(f(x)) = f^{-1}(x^2 - 5) \] Substitute \( x^2 - 5 \) into the function \( f^{-1}(x) \): \[ f^{-1}(x^2 - 5) = \sqrt{(x^2 - 5) + 5} \] Simplify: \[ \sqrt{(x^2 - 5) + 5} = \sqrt{x^2} = x \] Therefore, \( f^{-1}(f(x)) = x \).

Graphical Verification

Graphically, if two functions are inverses, their graphs should be symmetrical along the line \( y = x \). Plot the graphs of \( f(x) = x^2 - 5 \) (for \( x \geq 0 \)) and \( f^{-1}(x) = \sqrt{x+5} \). Check if they mirror each other along the line \( y = x \). This symmetry confirms that both functions are inverses.

Key concepts

function composition

To check if two functions are truly inverses of each other, we use function composition. This process involves combining one function with its supposed inverse. For Jocelyn and Gerry's function, we need to verify both compositions:

• Calculate if
  \(f(f^{-1}(x)) = x\)
• Calculate if
  \(f^{-1}(f(x)) = x\)

In essence, applying \(f^{-1}(x)\) to \(f(x)\) should return the original value \(x\), and vice versa. This shows that the functions undo each other, just like how adding and then subtracting the same number returns you to where you started. For instance, we found

• 
  \(f(f^{-1}(x)) = f(\root{x+5}) = (\root{x+5})^2-5 = x\)
• 
  Similarly, \(f^{-1}(f(x)) = \root{(x-5)+5} =\root{x^2} = x\)

By showing these equalities, we properly validate that the functions \(f(x)\) and \(f^{-1}(x)\) are indeed inverses of each other.

graphical verification

Graphical verification of inverse functions involves examining the symmetry of their graphs. If two functions are inverses, their graphs will be mirror images along the line \(y = x\). For the given functions, we plot \(f(x) = x^2 - 5\) and \(f^{-1}(x) = \root{x+5}\). When graphing, note the following:

• The graph of \(f(x) = x^2 - 5\) is a parabola starting from \( (0, -5) \) and opening upwards.
• 
  The graph of \(f^{-1}(x)=\root{x+5}\) is a square root function beginning at \(-5, 0\) and expanding to the right.

When plotting both, you'll notice they reflect over the line \(y = x\), confirming they are inverses. This method visually reassures the algebraic verification, giving a double-check via symmetry.

domain and range

Understanding the domain and range of functions is crucial, especially when dealing with inverses. The domain is the set of all possible inputs for the function, while the range is the set of all possible outputs.
For \(f(x) = x^2 - 5\), since we deal with non-negative values, its domain is \((x \text{ | } x \text{ ≥ } 0)\) and its range starts from -5 and goes to infinity: \((y \text{ | } y \text{ ≥ } -5)\).

The inverse \(f^{-1}(x) = \root{x+5}\) then reverses this: its domain is where the original function had its range, that is \((x \text{ | } x \text{ ≥ } -5)\), and its range covers non-negative outputs: \((y \text{ | } y \text{ ≥ } 0)\).This swap between domain and range is a hallmark of inverse functions. It ensures every input and output is appropriately paired and reinforces the consistency between the functions and their inverses.

square root function

The square root function, denoted as \(\root{x}\), is commonly encountered in inverse functions. For \(f^{-1}(x)= \root{x+5}\), it's essential to understand its characteristics:

• 
  It only accepts non-negative inputs (i.e., \(x \text{ ≥ } -5\).
• 
  Its output is always non-negative (i.e., \( y \text{ ≥ } 0\).

Graphically, the function starts from a specific point—in this case, \(-5, 0\)—and curves upward. This shape implies that for any non-negative output y, there’s a corresponding x-value.
Squaring a number is a straightforward process, but reversing it with a square root requires attention to the domain, as negative inputs would make the operation invalid or undefined.
This principle is essential when verifying if a function is an inverse, providing clarity and predictability to the function's behavior in different calculations.