Problem 63 - Algebra and Trigonometry

Question

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x\). \(g(x)=\sqrt{x+3}\)

Step-by-Step Solution

Answer

The inverse of the function is \( g^{-1}(x) = x^2 - 3 \).

Step 1: Understand the Function

The given function is \( g(x) = \sqrt{x+3} \). This function is a square root function that is defined for all \( x \geq -3 \) since square roots of negative numbers are not real.

Step 2: Set Function Equal to y

To find the inverse of the function, we first express the function in terms of \( y \). This gives us the equation:\[ y = \sqrt{x+3} \]

Step 3: Swap Variables x and y

In this step, swap \( x \) and \( y \) to begin finding the inverse:\[ x = \sqrt{y+3} \]

Step 4: Solve for y

Solve for \( y \) in terms of \( x \). Begin by squaring both sides to eliminate the square root:\[ x^2 = y + 3 \]Then, solve for \( y \) by subtracting 3 from both sides:\[ y = x^2 - 3 \]

Step 5: Write the Inverse Function

The inverse function, which we represent as \( g^{-1}(x) \), is:\[ g^{-1}(x) = x^2 - 3 \]

Step 6: Graph the Functions

Graph both \( g(x) = \sqrt{x+3} \) and its inverse \( g^{-1}(x) = x^2 - 3 \) on the same coordinate plane. Also, draw the line of reflection, \( y = x \), on the graph.

Step 7: Verify Graphical Reflection

Check to ensure that the graphs of \( g(x) \) and \( g^{-1}(x) \) are reflections across the line \( y = x \). This will verify that \( g^{-1} \) is correctly computed.

Key Concepts

One-to-One Function

A one-to-one function is essential for defining an inverse function. This means each input in the domain maps to a unique output in the codomain. If every distinct value of \( x \) leads to a distinct value of \( y \), then the function passes the "Horizontal Line Test."

In our example with \( g(x) = \sqrt{x+3} \), we recognize that this function is indeed one-to-one over its domain because:

• It is increasing for all \( x \geq -3 \).
• Each value of \( x+a \) exceeds \( x \), leading to \( g(x+a) > g(x) \).

By confirming it is one-to-one, we are assured that an inverse function exists for \( g(x) \). The computation of \( g^{-1}(x) \) only remains meaningful if this condition is satisfied.

Graphing Functions

Graphing is a crucial tool in visualizing mathematical relationships. It aids in understanding how changes in one variable affect another. When graphing a function and its inverse:

• Both \( g(x) \) and \( g^{-1}(x) \) should be graphed together on the same set of axes.
• For the function \( g(x) = \sqrt{x+3} \), the graph typically starts at the point \((-3, 0)\) and moves upward as \( x \) increases.
• The inverse, \( g^{-1}(x) = x^2 - 3 \), is a parabola opening upwards starting from \( y = -3 \).

By plotting these two functions, one can observe their relationship more concretely. This visualization assists in confirming the mathematical inversion done algebraically. It brings together analytical and graphical understanding of functions.

Reflection in the Line y=x

Understanding reflections in the line \( y=x \) is key to verifying inverse functions graphically. This line acts like a mirror, reflecting points equally between \( g(x) \) and \( g^{-1}(x) \). Here's how this works:

• Any point \((a, b)\) on \( g(x) \) transforms into \((b, a)\) on \( g^{-1}(x) \).
• The line \( y = x \) itself is a reference, serving as the mirror.
• When the two graphs are correctly reflected over \( y = x \), they demonstrate the function-inverse relationship visually.

In the given exercise, plotting \( g(x) = \sqrt{x+3} \) and \( g^{-1}(x) = x^2 - 3 \) alongside \( y = x \) lets us confirm their coordination and validate our inverse operation. The symmetry around \( y = x \) establishes that the inverse was calculated accurately. This adds a layer of visual proof to our earlier algebraic work.