Problem 67 - Intermediate Algebra

Question

Find the inverse of each function. Then graph the function and its inverse on one coordinate system. Show the line of symmetry on the graph. $$ f(x)=x^{2}-1(x \geq 0) $$

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = \sqrt{x + 1} \). Graph both functions and the line \( y = x \), showing symmetry.

Step 1: Understand the Function

The given function is \( f(x) = x^2 - 1 \) with the condition \( x \geq 0 \). This is a restricted domain of the standard quadratic function, ensures the function is one-to-one, and makes finding an inverse possible.

Step 2: Find the Inverse Function

To find the inverse, set \( y = f(x) = x^2 - 1 \). Therefore, we solve for \( x \) in terms of \( y \): \( y = x^2 - 1 \). Rearranging gives \( y + 1 = x^2 \). Taking the square root on both sides, we get \( x = \sqrt{y + 1} \). With \( x \geq 0 \), the positive square root is taken, thus the inverse function is \( f^{-1}(y) = \sqrt{y + 1} \).

Step 3: Sketch the Graph of the Function

Graph the original function \( f(x) = x^2 - 1 \) starting from the vertex at \( (0, -1) \) upwards. Since the domain is \( x \geq 0 \), only the right half of the parabola is drawn.

Step 4: Graph the Inverse Function

Graph the inverse function \( f^{-1}(x) = \sqrt{x + 1} \). This graph will start from the point \((-1, 0)\) and curve upwards to the right.

Step 5: Draw the Line of Symmetry

The line of symmetry between a function and its inverse is \( y = x \). Draw this line on the graph to show how the function \( f(x) = x^2 - 1 \) and its inverse \( f^{-1}(x) = \sqrt{x + 1} \) are reflections over this line.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function represented in the general form as \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants and \( a eq 0 \). These functions graph as a U-shaped curve called a parabola.
In the given exercise, we have a specific case of the quadratic function: \( f(x) = x^2 - 1 \). Here, the parabola is shifted down by one unit because of the \(-1\) in the equation. Since we consider only \( x \geq 0 \), we are dealing with a restricted domain.
Quadratic functions have important characteristics such as their vertex and axis of symmetry. In unrestricted parabolas, the axis of symmetry is a vertical line passing through the vertex of the parabola, specifically for our function, it is the y-axis \( x = 0 \). However, because of the restriction, the graph is only present for x-values greater than or equal to zero.

• Vertex: For \( f(x) = x^2 - 1 \), it is at \((0, -1)\).
• Direction: The parabola opens upwards since the coefficient of \( x^2 \) is positive.
• Domain and Range: With \( f(x) = x^2 - 1 \ : x \geq 0\), the domain is \( [0, \infty) \) and range is \( [-1, \infty) \).

Graphing Functions

Graphing functions involves representing mathematical equations visually. For the function \( f(x) = x^2 - 1 \), start by identifying key points like the vertex and intercepts, then plot them on a graph. Since the domain is restricted to \( x \geq 0 \), you will only draw the right side of the parabola. Begin at the vertex \((0, -1)\) and plot upwards to the right.
For the inverse function \( f^{-1}(x) = \sqrt{x + 1} \), graphing follows a similar approach. Convert input \( x \) into output using the inverse's formula and plot points starting from the lowest value of \((-1, 0)\). This inverse graph opens to the right.
When plotting both, clearly label each curve to avoid confusion.
Graphing tools, like graph paper or digital graphing software, enhance accuracy.

• Start from known points (like intercepts or vertex).
• Continue plotting, ensuring you follow the graph's direction and shape.
• Check that both functions are reflecting over the line of symmetry \( y = x \).

Symmetry in Functions

Symmetry is an essential concept when exploring functions and their inverses. It helps to graphically show how one function is transformed into its inverse.
In two-dimensional graphs, symmetry often refers to a reflection over a line. For functions and their inverses, this line of symmetry is \( y = x \), meaning any point \((a, b)\) on \( f(x) \) is mirrored as \((b, a)\) on \( f^{-1}(x) \).
Using symmetry in our specific function, we draw \( y = x \) on the graph to show this reflective property. The original function \( f(x) = x^2 - 1 \) is mirrored across this line to match its inverse \( f^{-1}(x) = \sqrt{x + 1} \).
This reflective symmetry is crucial in verifying that two graphs are indeed inverses. If the functions correctly reflect over \( y = x \), then one is the inverse of the other.

• The symmetry line \( y = x \) serves as a mirror.
• Each point on \( f(x) \) corresponds to a point on \( f^{-1}(x) \).
• It assures correctness of graphing inverses.

Understanding these principles deepens comprehension of inverse functions and aids in correctly sketching their graphical aspects.