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Chapter 8: Problem 29

In Exercises \(29-32,\) for each function \(Q\) find \(Q^{-1},\) if it exists. For those functions with inverses, find \(Q(3)\) and \(Q^{-1}(3)\). $$ Q(x)=\frac{2}{3} x-5 $$

Short Answer

Expert verified
The inverse is \(Q^{-1}(x) = \frac{3}{2}(x + 5)\). \(Q(3) = -3\) and \(Q^{-1}(3) = 12\).

Step by step solution

01

- Verify the Function is One-to-One

To find the inverse of a function, first verify that the function is one-to-one. This can be determined by checking that each value of the dependent variable is paired with exactly one value of the independent variable. Since the function is linear and its slope is not zero, it passes the Horizontal Line Test and is one-to-one.
02

- Replace Q(x) with y

To find the inverse function, start by replacing the function notation with y. So, rewrite the function as: $$ y = \frac{2}{3} x - 5 $$
03

- Solve for x in terms of y

Next, solve this equation for x. Start by isolating the term with x: $$ y + 5 = \frac{2}{3} x $$ Then, multiply each side by the reciprocal of \ \(\frac{2}{3}\ \): $$ x = \frac{3}{2} (y + 5) $$
04

- Replace y with x

To write the inverse function in standard function notation, replace y with x: $$ Q^{-1}(x) = \frac{3}{2}(x + 5) $$
05

- Evaluate Q(3)

Find the value of Q(3) by substituting 3 into the original function: $$ Q(3) = \frac{2}{3}(3) - 5 = 2 - 5 = -3 $$
06

- Evaluate Q^{-1}(3)

Find the value of the inverse function at 3 by substituting 3 into Q^{-1}(x): $$ Q^{-1}(3) = \frac{3}{2}(3 + 5) = \frac{3}{2} \times 8 = 12 $$

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Key Concepts

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

One-to-One Functions
Understanding one-to-one functions is crucial when dealing with inverse functions. A function is one-to-one if and only if it assigns a unique output to each input. This means that no two different input values map to the same output value. In simpler terms, every x-value has a unique y-value and vice versa.
This property is important because a function must be one-to-one to have an inverse. If a function is not one-to-one, its inverse would not be a function since it would assign more than one output to some inputs. You can check if a function is one-to-one by using the Horizontal Line Test.
Linear Functions
Linear functions are among the simplest and most common types of functions. They have the form:
\[ f(x) = mx + b \]
where 'm' is the slope and 'b' is the y-intercept. Linear functions are straight lines when graphed and have a constant rate of change (the slope).
In the given exercise, the function\ Q(x) = \frac{2}{3}x - 5 \ is a linear function. The slope 'm' here is \frac{2}{3} and the y-intercept 'b' is -5. Because the slope is not zero, the function is guaranteed to be one-to-one. Thus, this function can have an inverse.
Function Notation
Function notation is a way to denote functions in a standardized way. Instead of writing out equations, we use symbols to represent them. For example, we represent the function Q as \[ Q(x)= \frac{2}{3}x - 5 \]
This helps us manage and manipulate equations more efficiently, especially when dealing with multiple functions.
When finding an inverse function, we switch the roles of the dependent and independent variables. For instance, if Q maps x to a value, its inverse\ (Q^{-1})\ will map that value back to x.
Horizontal Line Test
The Horizontal Line Test is a visual method to check if a function is one-to-one. To apply this test, draw horizontal lines across the graph of the function. If any horizontal line intersects the graph more than once, then the function is not one-to-one.
SInce a one-to-one function must pass this test for an inverse to exist. For linear functions like \[ Q(x)= \frac{2}{3}x - 5 \], the graph is a straight line with a non-zero slope, ensuring it passes the Horizontal Line Test and confirming that the function is indeed one-to-one.

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Most popular questions from this chapter

If we know the radius and depth of a parabolic reflector, we also know where the focus is. a. Find a generic formula for the focal length \(f\) of a parabolic reflector expressed in terms of its radius \(R\) and depth \(D\). The focal length \(\left|\frac{1}{4 a}\right|\) is the distance between the vertex and the focal point. Assume \(a>0\). b. Under what conditions does \(f=D ?\)

On March \(2,2007,\) the conversion rate from U.S. dollars to euros was \(0.749 ;\) that is, on that day you could change \(\$ 1\) for 0.749 euros, the currency of the European Union. a. Was a U.S. dollar worth more or less than 1 euro? b. Using the March 2 exchange rate, construct a function \(C_{1}(d)\) that converts \(d\) dollars to euros. What is \(C_{1}(1) ?\) \(C_{1}(25) ?\) c. Now construct a second function \(\mathrm{C}_{2}(r)\) that converts \(r\) euros back to dollars. What is \(C_{2}(1) ? C_{2}(100) ?\) d. Show that \(C_{1}\) and \(C_{2}\) are inverses of each other. e. Reread the beginning of Exercise 8 , which describes a conversion process between Canadian and U.S. dollars. In that process the two formulas are not inverses of each other. Why not?

A shot-put athlete releases the shot at a speed of 14 meters per second, at an angle of 45 degrees to the horizontal (ground level). The height \(y\) (in meters above the ground) of the shot is given by the function $$ y=2+x-\frac{1}{20} x^{2} $$ where \(x\) is the horizontal distance the shot has traveled (in meters). a. What was the height of the shot at the moment of release? b. How high is the shot after it has traveled 4 meters horizontally from the release point? 16 meters? c. Find the highest point reached by the shot in its flight. d. Draw a sketch of the height of the shot and indicate how far the shot is from the athlete when it lands.

The formula for the volume of a cone is \(V=\left(\frac{1}{3}\right) \pi r^{2} h .\) Assume you are holding a 6 -inch-high sugar cone for ice cream. a. Construct a function \(V(r)\) for the volume as a function of \(r\). Why don't you need the variable \(h\) in this case? Find \(V(1.5)\) and explain what have you found (using appropriate units). b. Evaluate \(V^{-1}(25) .\) Describe your results. What are the units attached to the number \(25 ?\) c. When dealing with abstract functions where \(f(x)=y\), we have sometimes used the convention of using \(x\) (rather than \(y\) ) as the input to the inverse function \(f^{-1}(x)\). Explain why it does not make sense to interchange \(V\) and \(r\) here to find the inverse function.

a. Construct a quadratic function with zeros at \(x=1\) and \(x=2 .\) b. Is there more than one possible quadratic function for part (a)? Why or why not?
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