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Chapter 3: Problem 61

A function \(f\) is given. (a) Sketch the graph of \(f .(b)\) Use the graph of \(f\) to sketch the graph of \(f^{-1} .(c)\) Find \(f^{-1} .\) $$ f(x)=3 x-6 $$

Short Answer

Expert verified
The inverse function is \(f^{-1}(x) = \frac{x+6}{3}\).

Step by step solution

01

Understand the Given Function

The function given is a linear function, i.e., a straight line: \( f(x) = 3x - 6 \). This line has a slope of 3 and a y-intercept at \(-6\).
02

Sketch the Graph of f(x)

To sketch \( f(x) = 3x - 6 \), start by plotting the y-intercept (0, -6) on the Cartesian plane. Then, from this point, use the slope of 3 (which means rise 3 and run 1) to find another point, e.g., (1, -3). Connect these points to form the line.
03

Concept of Inverse Function

To sketch the graph of \( f^{-1} \), we need to reflect the graph of \( f(x) \) over the line \( y = x \). The inverse function will pass through the points that are swaps of the points on \( f(x) \).
04

Sketch the Graph of f^{-1}(x)

Swap the coordinates of the points on \( f(x) \). For example, if \( f(x) \) passes through (0, -6), \( f^{-1}(x) \) will pass through (-6, 0). Use another point such as (1, -3) becoming (-3, 1). Reflect the line across \( y = x \).
05

Find the Inverse Function f^{-1}(x)

To find \( f^{-1}(x) \), solve the equation \( y = 3x - 6 \) for \( x \). \( y = 3x - 6 \Rightarrow y + 6 = 3x \Rightarrow x = \frac{y+6}{3} \). Thus, \( f^{-1}(x) = \frac{x+6}{3} \).

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Key Concepts

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

Linear Functions
Linear functions are among the most basic types of mathematical functions you will encounter. Their defining characteristic is their straight-line graph. A linear function can generally be written in the form:
  • \( f(x) = mx + b \)
where \( m \) represents the slope, and \( b \) represents the y-intercept. In our exercise, the function given is \( f(x) = 3x - 6 \). This equation tells us a few important things:
  • The slope \( m = 3 \) indicates how steep the line is and the direction it goes. A positive slope means the line rises as it moves from left to right.
  • The y-intercept \( b = -6 \) tells us where the line crosses the y-axis. The graph will pass through the point (0, -6).
By understanding these elements, we can easily sketch or interpret linear functions.
Graphing Functions
Graphing a function like \( f(x) = 3x - 6 \) involves plotting points and connecting them into a straight line. Here's how you can do it:
  • Start by plotting the y-intercept at the point (0, -6). This is where the line will intersect with the y-axis.
  • Use the slope of the function to find another point. The slope of 3 can be interpreted as "rise over run," meaning for every 3 units you move up, you move 1 unit to the right. From (0, -6), moving up 3 units and right 1 unit gives you the point (1, -3).
  • Join these two points with a straight line, extending it in both directions to cover the graph.
This is the complete graph of the linear function \( f(x) = 3x - 6 \). These steps can be easily followed with any linear function by just updating the slope and y-intercept values as needed.
Function Reflection
Understanding inverse functions involves the concept of reflecting a graph over the line \( y = x \). The inverse function, denoted as \( f^{-1}(x) \), essentially swaps the x and y values of the original function \( f(x) \).
  • To sketch an inverse, reflect points on the graph of \( f(x) \) over the line \( y = x \).
  • Pretending the line \( y=x \) is a mirror, points like (0, -6) on \( f(x) \) would reflect to (-6, 0) on \( f^{-1}(x) \).
  • This method works because, for an inverse function, the input and output are switched.
To find the equation of the inverse function of \( f(x) = 3x - 6 \), solve for \( x \) in terms of \( y \):
  • Start with \( y = 3x - 6 \).
  • Rearrange to get \( 3x = y + 6 \).
  • Solve for \( x \) to get \( x = \frac{y + 6}{3} \).
Thus, the inverse function is \( f^{-1}(x) = \frac{x + 6}{3} \), completing the reflection process.

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Most popular questions from this chapter

Multiple Discounts \(A\) car dealership advertises a 15\(\%\) discount on all its new cars. In addition, the manufacturer offers a \(\$ 1000\) rebate on the purchase of a new car. Let \(x\) represent the sticker price of the car. (a) Suppose only the 15\(\%\) discount applies. Find a function \(f\) that models the purchase price of the car as a function of the sticker price \(x .\) (b) Suppose only the \(\$ 1000\) rebate applies. Find a function \(g\) that models the purchase price of the car as a function of the sticker price \(x\) (c) Find a formula for \(H=f \circ g .\) (d) Find \(H^{-1} .\) What does \(H^{-1}\) represent? (e) Find \(H^{-1}(13,000) .\) What does your answer represent?

The given function is not one-to-one. Restrict its domain so that the resulting function \(i s\) one-to-one. Find the inverse of the function with the restricted domain. (There is more than one correct answer.) $$ h(x)=(x+2)^{2} $$

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=\frac{x}{x+1}, \quad g(x)=\frac{1}{x} $$

Find \(f+g, f-g \cdot f g,\) and \(f / g\) and their domains. $$ f(x)=\frac{2}{x+1}, \quad g(x)=\frac{x}{x+1} $$

Area of a Ripple \(A\) stone is dropped in a lake, creating a circular ripple that travels outward at a speed of 60 \(\mathrm{cm} / \mathrm{s}\) . (a) Find a function \(g\) that models the radius as a function of time. (b) Find a function \(f\) that models the area of the circle as a function of the radius. (c) Find \(f \circ g .\) What does this function represent?
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