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Chapter 1: Problem 8

In Exercises \(7-12,\) determine whether the function has an inverse function. $$y=x^{2}+5 x$$

Short Answer

Expert verified
The function \(y = x^2 + 5x\) does have an inverse, because it is a one-to-one function.

Step by step solution

01

Find the derivative

The derivative of the function will help to identify the increasing and decreasing intervals of the function, which is crucial in figuring out if it's one-to-one. The derivative of \(y = x^2 + 5x\) is \(y' = 2x + 5\).
02

Analyze the derivative

Set \(y' = 0\) and solve for \(x\). Using \(2x + 5 = 0\), x=-5/2. This is the x-coordinate where the function changes from increasing to decreasing or vice versa. For \(x < -5/2\), the function decreases, and for \(x > -5/2\), the function increases. Since it doesn't decrease and increase simultaneously in any interval, we conclude that it only exists once in every value of \(y\).
03

Apply the horizontal line test

The horizontal line test is a theoretical test that involves imagining drawing a horizontal line through the graph of the function. If at any point the line crosses the graph more than once, then the function does not have an inverse. For this function, any horizontal line will only intersect the graph once - so the function is one-to-one and thus does have an inverse.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Derivatives
Derivatives are essential in calculus and help us understand how functions change. When you take the derivative of a function, you're essentially looking at its rate of change at any given point. For example, if you have a function like \(y = x^2 + 5x\), as in your exercise, the derivative is \(y' = 2x + 5\). This tells us how the function's output, \(y\), changes as \(x\) changes.
Here, the derivative \(y' = 2x + 5\) is a linear function, which gives us important information:
  • When \(x < -5/2\), the derivative is negative, indicating that the function is decreasing in that interval.
  • When \(x > -5/2\), the derivative is positive, indicating that the function is increasing.
By understanding the derivative, we can analyze the behavior of the function and determine the intervals where it increases or decreases.
Exploring One-to-One Functions
A one-to-one function has a special characteristic: it assigns each input exactly one unique output, and vice versa. To have an inverse, a function must be one-to-one, meaning it must pass the horizontal line test. In mathematical terms, for every value of \(y\), there should be exactly one corresponding \(x\).
In your exercise, the function \(y = x^2 + 5x\) is analyzed using its derivative. It was found to be decreasing for \(x < -5/2\) and increasing for \(x > -5/2\). This behavior suggests that as \(x\) changes, \(y\) doesn't repeat values, ensuring that each \(y\) is paired with a single \(x\). Therefore, this function meets the criteria to be one-to-one.
The Horizontal Line Test
The horizontal line test is a visual way to determine if a function has an inverse. If every horizontal line drawn through the graph of a function intersects the curve at most once, then the function is one-to-one and has an inverse.
For the function \(y = x^2 + 5x\), you can imagine drawing horizontal lines across the graph:
  • If any horizontal line touches the graph in more than one place, the function would not be one-to-one.
  • However, for this function, each horizontal line only touches at one point, confirming that the function is indeed one-to-one.
This simple yet effective test reassures us that the function can be inverted due to its one-to-one nature.

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Most popular questions from this chapter

Multiple Choice John invests \(\$ 200\) at 4.5\(\%\) compounded annually. About how long will it take for John's investment to double in value? (A) 6 yrs (B) 9 yrs (C) 12 yrs (D) 16 yrs (E) 20 yrs

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Population of Texas Table 1.11 gives the population of Texas for several years. Population of Texas $$\begin{array}{ll}{\text { Year }} & {\text { Population (thousands) }} \\\ {1980} & {14,229} \\ {1990} & {16,986} \\ {1995} & {18,159} \\ {1998} & {20,158} \\ {1999} & {20,558} \\ {2000} & {20,852}\end{array}$$ (a) Let \(x=0\) represent \(1980, x=1\) represent \(1981,\) and so forth. Find an exponential regression for the data, and superimpose its graph on a scatter plot of the data. (b) Use the exponential regression equation to estimate the population of Texas in \(2003 .\) How close is the estimate to the actual population of \(22,119,000\) in 2003\(?\) (c) Use the exponential regression equation to estimate the annual rate of growth of the population of Texas.

Group Activity Inverse Functions Let \(y=f(x)=m x+b\) \(m \neq 0\) (a) Writing to Learn Give a convincing argument that \(f\) is a one-to-one function. (b) Find a formula for the inverse of \(f .\) How are the slopes of \(f\) and \(f^{-1}\) related? (c) If the graphs of two functions are parallel lines with a nonzero slope, what can you say about the graphs of the inverses of the functions? (d) If the graphs of two functions are perpendicular lines with a nonzero slope, what can you say about the graphs of the inverses of the functions? .
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