Problem 78
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). $$ f(x)=\frac{x^{3}}{2} $$
Step-by-Step Solution
Verified Answer
The function \(f(x) = \frac{x^{3}}{2}\) has an inverse that is a function, thus is a one-to-one function.
1Step 1: Plot the Function
Using a graphing utility, plot the function \(f(x) = \frac{x^{3}}{2}\)
2Step 2: Analyze the Graph
By looking at the curve, see if the function is one-to-one. This can be visually confirmed if no horizontal line intersects the curve more than once. This is known as the Horizontal Line Test.
3Step 3: Determine if Function has Inverse
If the function passes the Horizontal Line Test, it has an inverse that is a function. This means no horizontal line intersected the curve more than once, which indicates a one-to-one function
Key Concepts
Inverse FunctionsHorizontal Line TestGraphing Utilities
Inverse Functions
An inverse function is a special type of function that essentially reverses the effect of the original function. If you have a function, let's call it \( f(x) \), and you apply it to an input \( x \), you'll get an output \( y \). The inverse function does the opposite: apply it to \( y \), and you should get back your original input \( x \).
For a function to have an inverse that is also a function, each output \( y \) must come from exactly one input \( x \), making it a one-to-one function. This is crucial because if a function is not one-to-one, its inverse will not pass the function test: a test that checks if each input has a unique output.
When dealing with inverse functions, always remember that not all functions have inverses that are functions. Consider using visual tests like the Horizontal Line Test to determine if a function is indeed one-to-one, signaling that it can have an inverse.
For a function to have an inverse that is also a function, each output \( y \) must come from exactly one input \( x \), making it a one-to-one function. This is crucial because if a function is not one-to-one, its inverse will not pass the function test: a test that checks if each input has a unique output.
When dealing with inverse functions, always remember that not all functions have inverses that are functions. Consider using visual tests like the Horizontal Line Test to determine if a function is indeed one-to-one, signaling that it can have an inverse.
Horizontal Line Test
The Horizontal Line Test is a simple yet powerful way to determine if a function is one-to-one. It involves an imaginary horizontal line that runs across the graph of the function. To perform this test, imagine drawing several horizontal lines across the graph. If none of these lines intersect the graph more than once, then the function is one-to-one.
In simpler terms, no two different input values \( x eq x' \) should map to the same output value \( y \). This ensures each output corresponds to only one input. Functions that pass the Horizontal Line Test can have inverses that are functions.
In our example function \( f(x) = \frac{x^3}{2} \), a graph would show the curve continually increasing, meaning any horizontal line drawn across it would intersect only once. Hence, it passes the Horizontal Line Test, verifying that the inverse exists and is a function.
In simpler terms, no two different input values \( x eq x' \) should map to the same output value \( y \). This ensures each output corresponds to only one input. Functions that pass the Horizontal Line Test can have inverses that are functions.
In our example function \( f(x) = \frac{x^3}{2} \), a graph would show the curve continually increasing, meaning any horizontal line drawn across it would intersect only once. Hence, it passes the Horizontal Line Test, verifying that the inverse exists and is a function.
Graphing Utilities
Graphing utilities are incredibly helpful tools that assist in visualizing functions and performing tests like the Horizontal Line Test. Today, many software options and calculator devices provide this capability, making it accessible for students and educators alike.
By using graphing utilities, you can plot complex functions that may be difficult to visualize by hand. For example, you can use software to easily plot \( f(x) = \frac{x^3}{2} \) and automatically identify if the curve suggests the function is one-to-one.
Some benefits of graphing utilities include:
By using graphing utilities, you can plot complex functions that may be difficult to visualize by hand. For example, you can use software to easily plot \( f(x) = \frac{x^3}{2} \) and automatically identify if the curve suggests the function is one-to-one.
Some benefits of graphing utilities include:
- Easy and quick visualization of functions.
- Ability to test functions for one-to-one characteristics using visual aids.
- Helpful when dealing with higher complexity mathematical problems.
Other exercises in this chapter
Problem 77
Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\sqrt{x+2}-2 $$
View solution Problem 77
What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.
View solution Problem 78
Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\sqrt{x+1}-1 $$
View solution Problem 78
Give an example of a circle's equation in standard form. Describe how to find the center and radius for this circle.
View solution