Problem 76
Question
Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$f(x)=x^{2}-1$$
Step-by-Step Solution
Verified Answer
The function \(f(x)=x^{2}-1\) does not have an inverse that is a function, as it is not one-to-one.
1Step 1: Graph the Function
Use a graphing utility to plot the given function \(y = x^{2}-1\). The graph is a parabola that opens upward, with its vertex at (0, -1).
2Step 2: Apply the Horizontal Line Test
Draw horizontal lines on the graph. If any of these lines intersect the graph of the function at more than one point, then the function is not one-to-one.
3Step 3: Determine if the Function has an Inverse
From the horizontal lines you drew in Step 2, you can see they intersect the graph at more than one point. Therefore, the function \(f(x)=x^{2}-1\) is not one-to-one and does not have an inverse that is a function.
Key Concepts
Horizontal Line TestGraphing UtilityOne-to-One FunctionParabolas
Horizontal Line Test
The horizontal line test is a simple graphical check to determine if a function has an inverse that is also a function. To perform the test, you draw multiple horizontal lines at different y-values across the graph of the function. If any horizontal line intersects the graph more than once, then the function is not one-to-one, meaning it does not have an inverse that is also a function.
This test is particularly useful because it provides a quick visual cue. When applying it to parabolas, such as the given function \f\((x)=x^{2}-1\f\), you'll notice that any horizontal line above y=-1 intersects the graph twice, indicating that the original function isn't one-to-one and thus doesn't have a function as its inverse.
This test is particularly useful because it provides a quick visual cue. When applying it to parabolas, such as the given function \f\((x)=x^{2}-1\f\), you'll notice that any horizontal line above y=-1 intersects the graph twice, indicating that the original function isn't one-to-one and thus doesn't have a function as its inverse.
Graphing Utility
A graphing utility is an essential tool in mathematics that helps visualize functions. It can be as simple as a graphing calculator or as advanced as computer software. When you input a function, the utility plots it on a coordinate grid.
For the exercise involving the parabola \f\((x)=x^{2}-1\f\), a graphing utility gives a clear visual of the shape and position of the graph. With this tool, users can also zoom in and out or adjust the window settings to better understand the behavior of the function across different intervals. Always keep in mind that while graphing utilities are powerful, interpreting the graph correctly is still up to the user.
For the exercise involving the parabola \f\((x)=x^{2}-1\f\), a graphing utility gives a clear visual of the shape and position of the graph. With this tool, users can also zoom in and out or adjust the window settings to better understand the behavior of the function across different intervals. Always keep in mind that while graphing utilities are powerful, interpreting the graph correctly is still up to the user.
One-to-One Function
A function is considered one-to-one if each input is associated with a unique output, and no two different inputs produce the same output. This uniqueness makes it possible to 'reverse' the function, finding an inverse function where the roles of inputs and outputs are swapped.
When a function like \f\((x)=x^{2}-1\f\) fails to be one-to-one, it means there's a certain y-value that can come from more than one x-value. For instance, both x=2 and x=-2 give a y-value of 3, therefore, the function is not one-to-one, and its inverse is not a function. Understanding the concept of one-to-one functions is crucial when exploring the invertibility of functions.
When a function like \f\((x)=x^{2}-1\f\) fails to be one-to-one, it means there's a certain y-value that can come from more than one x-value. For instance, both x=2 and x=-2 give a y-value of 3, therefore, the function is not one-to-one, and its inverse is not a function. Understanding the concept of one-to-one functions is crucial when exploring the invertibility of functions.
Parabolas
Parabolas are U-shaped graphs that can open upwards or downwards and are the result of graphing quadratic functions. The general form of a quadratic function is \f\((x) = ax^{2} + bx + c\f\), where a, b, and c are constants.
The vertex of the parabola represents either the maximum or minimum point, depending on its direction. For the function \f\((x)=x^{2}-1\f\), the vertex is at (0, -1), and the parabola opens upwards. This specific shape means that for every positive y-value, there are two x-values that correspond to it, further highlighting why the function is not one-to-one.
The vertex of the parabola represents either the maximum or minimum point, depending on its direction. For the function \f\((x)=x^{2}-1\f\), the vertex is at (0, -1), and the parabola opens upwards. This specific shape means that for every positive y-value, there are two x-values that correspond to it, further highlighting why the function is not one-to-one.
Other exercises in this chapter
Problem 76
In your own words, describe how to find the midpoint of a line segment if its endpoints are known.
View solution Problem 76
Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=2 x^{2}$$
View solution Problem 76
Express the given function \(h\) as a composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=(2 x-5)^{3}$$
View solution Problem 76
Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$g(x)=2 \sqrt{x+1}$$
View solution