Problem 68 - Elementary and 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 function is \(f^{-1}(x) = \sqrt{x - 1}\).

Step 1: Find the inverse function

To find the inverse function of \(f(x) = x^2 + 1\) where \(x \geq 0\), we start by replacing \(f(x)\) with \(y\):\[ y = x^2 + 1 \]Switch \(x\) and \(y\) to find the inverse, getting:\[ x = y^2 + 1 \]Solve for \(y\) by isolating the term with \(y\):\[ x - 1 = y^2 \]Taking the square root of both sides, and since \(x \geq 0\), we consider the positive root:\[ y = \sqrt{x - 1} \]Thus, the inverse function is \(f^{-1}(x) = \sqrt{x - 1}\).

Step 2: Graph both functions

Graph the original function \(f(x) = x^2 + 1\) which is a parabola opening upwards starting from the point (0,1).Next, graph the inverse function \(f^{-1}(x) = \sqrt{x - 1}\), which is a square root function starting from the point (1,0) and opening to the right, as \(x\) must be greater than or equal to 1 to keep the expression under the square root non-negative.

Step 3: Draw the line of symmetry

The line of symmetry when graphing a function and its inverse is the line \(y = x\).On the coordinate system, draw the line \(y = x\). This line reflects each point of the original function to the inverse function and vice versa.

Key Concepts

Graphing Functions

Graphing is a simple way to visually represent a function. It helps us see the relationship between two variables, usually x (horizontal axis) and y (vertical axis). When graphed, every point on a curve or line represents a pair of values that satisfy the equation of the function. This provides a clear picture of how the function behaves. For quadratic functions like our example, the graph is a parabola.

• The vertex of the parabola is a crucial point. In our case, it is at (0, 1) for the function \(f(x) = x^2 + 1\).
• The graph's shape tells us that the function is increasing as x moves away from zero.
• The domain of the original function is limited to \(x \geq 0\), ensuring that we only consider the right side of the parabola.

Understanding how to graph functions allows us to visualize important characteristics such as intercepts, turning points, and how they interact with their inverses.

Lines of Symmetry

Lines of symmetry in graphs represent axes where the graph can be folded, and both sides match perfectly. For functions and their inverses, the line \(y = x\) serves as this fold line or reflection line.

• It is an important tool for checking if two graphs are inverses.
• This line shows the reflection property of a function and its inverse.
• If the graph of one function can be flipped over this line to match another, it confirms their inverse relationship.

In our exercise, drawing the line \(y = x\) helps verify that the graphs of \(f(x) = x^2 + 1\) and \(f^{-1}(x) = \sqrt{x - 1}\) are reflections of each other across this symmetry line. Thus, the line of symmetry is not only a geometric concept but a tool for understanding functions intuitively.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, with a general form of \(f(x) = ax^2 + bx + c\). They produce a U-shaped curve known as a parabola when graphed.
Some key properties of quadratic functions to keep in mind include:

• Vertex: The highest or lowest point of the parabola. In our case, the vertex is (0, 1).
• Axis of symmetry: A vertical line that runs through the vertex, dividing the parabola into two mirror images.
• Direction: The parabola opens upwards if \(a > 0\) (as it does in our function \(f(x) = x^2 + 1\)), or downwards if \(a < 0\).

Quadratic functions have real-world applications in physics, engineering, finance, and more. Understanding their graphs and behaviors is essential for solving complex mathematical problems.