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Chapter 0: Problem 3

In each part, use the horizontal line test to determine whether the function f is one-to-one. $$ \begin{array}{ll}{\text { (a) } f(x)=3 x+2} & {\text { (b) } f(x)=\sqrt{x-1}} \\\ {\text { (c) } f(x)=|x|} & {\text { (d) } f(x)=x^{3}} \\ {\text { (e) } f(x)=x^{2}-2 x+2} & {\text { (f) } f(x)=\sin x}\end{array} $$

Short Answer

Expert verified
(a) Yes, (b) Yes, (c) No, (d) Yes, (e) No, (f) No.

Step by step solution

01

Understanding the Horizontal Line Test

The horizontal line test is used to determine if a function is one-to-one. A function is one-to-one if and only if no horizontal line intersects the graph of the function more than once.
02

Analyze Function (a) - Linear Function

Given function: \( f(x) = 3x + 2 \). A linear equation of the form \( f(x) = mx + b \) where \( m eq 0 \) is always one-to-one. Any horizontal line will intersect the graph at most once.
03

Analyze Function (b) - Square Root Function

Given function: \( f(x) = \sqrt{x-1} \). A square root function with the domain \( x \geq 1 \) (to ensure the square root is defined for real numbers) is one-to-one. Horizontal lines intersect it at most once.
04

Analyze Function (c) - Absolute Value Function

Given function: \( f(x) = |x| \). This function is not one-to-one because horizontal lines can intersect it in two places. For example, \( f(x) = |1| = |鈭�1| = 1 \).
05

Analyze Function (d) - Cubic Function

Given function: \( f(x) = x^3 \). The cubic function is one-to-one because it passes the horizontal line test. Any horizontal line will intersect the graph of the function at most once.
06

Analyze Function (e) - Quadratic Function

Given function: \( f(x) = x^2 - 2x + 2 \). This is a quadratic function which is a parabola opening upwards and not one-to-one, as horizontal lines can intersect it at two points.
07

Analyze Function (f) - Sine Function

Given function: \( f(x) = \sin x \). This function is periodic and not one-to-one. Horizontal lines intersect it infinitely many times over its range.

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Key Concepts

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

One-to-One Functions
One-to-one functions are special kinds of functions where each input value maps to a unique output value. In simpler terms, no two different inputs will give you the same output. This can be checked using the horizontal line test. If a horizontal line crosses the graph of the function more than once, the function is not one-to-one. This is crucial because one-to-one functions have an inverse that is also a function. Inverses are essential in many fields, such as solving equations and understanding geometric transformations.
Linear Functions
Linear functions are among the simplest types of functions and are represented by the equation \( f(x) = mx + b \). Here, \( m \) is the slope and \( b \) is the y-intercept. These functions graph as straight lines and are always one-to-one if \( m eq 0 \). This is because any horizontal line drawn across the graph of a linear function will intersect it at most once. Linear functions are ubiquitous in algebra and are fundamental to understanding other function types.
  • Simple to understand and visualize.
  • Always pass the horizontal line test if not horizontal (i.e., \( m eq 0 \)).
Quadratic Functions
Quadratic functions are represented by the function \( f(x) = ax^2 + bx + c \). These functions graph as parabolas. A standard quadratic function does not pass the horizontal line test since horizontal lines can intersect them at multiple points. This means they are not one-to-one. These functions are used to describe many real-world phenomena, such as projectile motion or the shape of a parabolic dish. Their graphs can open upwards or downwards based on the sign of \( a \).
  • Parabolas open either up or down.
  • Typically not one-to-one unless restricted to half its domain.
Cubic Functions
Cubic functions take the form \( f(x) = ax^3 + bx^2 + cx + d \). These functions produce an S-shaped curve when plotted on a graph, with one part increasing and another decreasing. Unlike quadratic functions, cubic functions pass the horizontal line test and are one-to-one. This means they have a unique output for every input. Cubic functions are useful in calculus and modeling situations where the variable grows at different rates.
  • Graph has an S-shape, crossing the x-axis up to three times.
  • Always one-to-one, reflecting complex, dynamic changes in variables.

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Most popular questions from this chapter

Determine whether the statement is true or false. Explain your answer. A one-to-one function is invertible.

(a) Make a conjecture about the general shape of the graph of \(y=\log (\log x),\) and sketch the graph of this equation and \(y=\log x\) in the same coordinate system. (b) Check your work in part (a) with a graphing utility.

(a) Prove: If \(f\) and \(g\) are one-to-one, then so is the composition \(f \circ g\). (b) Prove: If \(f\) and \(g\) are one-to-one, then $$ (f \circ g)^{-1}=g^{-1} \circ f^{-1} $$

Solve for \(x\) without using a calculating utility. $$ \ln (1 / x)=-2 $$

$$ \begin{array}{l}{\text { Prove: }} \\ {\text { (a) } \sin ^{-1} x=\tan ^{-1} \frac{x}{\sqrt{1-x^{2}}}(|x|<1)} \\ {\text { (b) } \cos ^{-1} x=\frac{\pi}{2}-\tan ^{-1} \frac{x}{\sqrt{1-x^{2}}} \quad(|x|<1)}\end{array} $$
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