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Chapter 2: Problem 90

For the linear function \(f(x)=m x+b\) to be one-to-one, what must be true about its slope? If it is one-to-one, find its inverse. Is the inverse linear? If so, what is its slope?

Short Answer

Expert verified
For one-to-one, \(m \neq 0\). The inverse is linear with slope \(\frac{1}{m}\).

Step by step solution

01

Understanding One-to-One Functions

A function is one-to-one if each output value corresponds to exactly one input value. For linear functions in the form of \(f(x) = mx + b\), this means the graph must either constantly increase or constantly decrease, which occurs when the slope \(m\) is non-zero.
02

Determine Slope Condition

Since a one-to-one function must be always increasing or decreasing, \(m\) (the slope) must be different from zero. If \(m=0\), the function is constant and not one-to-one.
03

Find the Inverse Function

To find the inverse of \(f(x) = mx + b\), swap \(x\) and \(y\) in the equation and solve for \(y\):1. Start with: \(y = mx + b\).2. Swap \(x\) and \(y\): \(x = my + b\).3. Solve for \(y\): \(x - b = my\) \(y = \frac{x - b}{m}\).Thus, the inverse function is \(f^{-1}(x) = \frac{x - b}{m}\).
04

Determine If Inverse is Linear

The inverse function \(f^{-1}(x) = \frac{x - b}{m}\) is linear because it can be expressed in the form of \(y = mx + c\) as \(y = \frac{1}{m}x - \frac{b}{m}\), which is a linear equation.
05

Find the Slope of the Inverse Function

The slope of the inverse function \(f^{-1}(x) = \frac{x-b}{m}\) is \(\frac{1}{m}\). This is the coefficient of \(x\) in the inverse equation.

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Key Concepts

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

Linear Function
A linear function is a basic type of function in mathematics, defined by the formula \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This function is called "linear" because it graphs as a straight line.

A key feature of a linear function is how it behaves based on its slope, \( m \). If the slope is non-zero, the function increases (if \( m > 0 \)) or decreases (if \( m < 0 \)) consistently, making it one-to-one.
  • A one-to-one function ensures each output is unique to one input, essential for certain aspects of mathematics like finding inverses.
  • For a linear function to be one-to-one, its slope must be non-zero.
Understanding these properties is crucial as it directly impacts whether the function can have an inverse.
Inverse Function
The inverse function essentially "reverses" what the original function does. For a linear function \( f(x) = mx + b \), the inverse is found by swapping the \( x \) and \( y \) in the equation and solving for \( y \).

Here鈥檚 a simple step-by-step for finding the inverse:
  • Start with \( y = mx + b \).
  • Swap \( x \) and \( y \) to get \( x = my + b \).
  • Solve for \( y \) to get \( y = \frac{x - b}{m} \).
The result, \( f^{-1}(x) = \frac{x - b}{m} \), is the inverse function.

This new function is also linear because it maintains the form \( y = ax + c \), where \( a \) represents the new slope. Finding inverses is useful for reversing operations and understanding symmetric mathematical relationships.
Slope of a Line
The slope of a line is a measure of its steepness and direction. In any linear equation like \( f(x) = mx + b \), \( m \) is the slope. It determines how "steep" the line is and whether it ascends or descends:
  • A positive slope (\( m > 0 \)) means the line ascends from left to right.
  • A negative slope (\( m < 0 \)) means the line descends from left to right.
  • A zero slope (\( m = 0 \)) indicates a horizontal line, which does not qualify as one-to-one.
For an inverse function, the slope is important as well. If the original function's slope is \( m \), then the inverse function's slope is \( \frac{1}{m} \).

This reciprocal relationship is key in calculating how inputs relate to outputs oppositely, ensuring an accurate function inversion.

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Most popular questions from this chapter

Express the rule in function notation. (For example, the rule "square, then subtract 5 " is expressed as the function \(f(x)=x^{2}-5\) Take the square root, add \(8,\) then multiply by \(\frac{1}{3}\)

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\) $$f(x)=5$$

Evaluate the function at the indicated values. $$\begin{aligned} &f(x)=x^{2}+2 x ; \\ &f(0), f(3), f(-3), f(a), f(-x), f\left(\frac{1}{a}\right) \end{aligned}$$

A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$f(x)=x^{2 / 5}$$

Use the function to evaluate the indicated expressions and simplify. $$f(x)=x^{2}+1 ; \quad f(x+2), f(x)+f(2)$$
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