Problem 14 Sketch a graph of a function tha... [FREE SOLUTION]

Chapter 1: Problem 14

Sketch a graph of a function that is one-to-one on the intervals $(-\infty,-2),$ and $(0, \infty)$ but is not one-to-one on $(-\infty, \infty)$.

Short Answer

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Question: Sketch a graph of a function that is one-to-one on the intervals $(-\infty, -2)$ and $(0, \infty)$, but not one-to-one on $(-\infty, \infty)$. Answer: Define the piecewise function $f(x)$ as follows: $$ f(x) = \begin{cases} x^3 & x < -2 \\ \frac{x}{4} - \frac{1}{4} & -2 \le x \le 0 \\ x^2 & 0 < x \\ \end{cases} $$ To sketch the graph, draw the graph of $x^3$ in the interval $(-\infty, -2)$, the line segment $\displaystyle\frac{x}{4} - \displaystyle\frac{1}{4}$ on the interval $[-2, 0]$, and the graph of $x^2$ in the interval $(0, \infty)$. Connect the segments smoothly to form a single graph. This graph represents a function that is one-to-one on the intervals $(-\infty, -2)$ and $(0, \infty)$, but not one-to-one on $(-\infty, \infty)$.

Step by step solution

Define the piecewise function

Let's define a piecewise function $f(x)$ which combines two one-to-one functions, for example, $x^3$ for $(-\infty, -2)$ and $x^2$ for $(0, \infty)$. We will have: $$ f(x) = \begin{cases} x^3 & x < -2 \\ x^2 & 0 < x \\ \end{cases} $$

Connect the two intervals

Now we need to connect the intervals $(-\infty, -2)$ and $(0, \infty)$. We will use a straight line that intersects the graphs between those intervals to break the one-to-one property. In this case, let's choose a line segment $\displaystyle\frac{x}{4} - \displaystyle\frac{1}{4}$ on $[-2,0]$. The constant $-\displaystyle\frac{1}{4}$ ensures that the slope $\displaystyle\frac{1}{4}$ intersects the two functions at the desired points. The extended piecewise function becomes: $$ f(x) = \begin{cases} x^3 & x < -2 \\ \frac{x}{4} - \frac{1}{4} & -2 \le x \le 0 \\ x^2 & 0 < x \\ \end{cases} $$

Sketch the graph

To sketch the graph of $f(x)$, first draw the graph of $x^3$ in the interval $(-\infty, -2)$. Then, sketch the line segment $\displaystyle\frac{x}{4} - \displaystyle\frac{1}{4}$ on the interval $[-2,0]$. Finally, sketch the graph of $x^2$ in the interval $(0, \infty)$. Connect the segments smoothly to form a single graph. The resulting graph represents a function that is one-to-one on the intervals $(-\infty, -2)$ and $(0, \infty)$ but is not one-to-one on the entire real line $(-\infty, \infty)$ because it has duplicate y-values within the line segment.

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Key Concepts

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

One-to-One Functions

A function is known as one-to-one if every element of the function's domain maps to a unique element in its range. This means that no two different inputs will produce the same output.

Checking if a function is one-to-one involves a horizontal line test. If any horizontal line crosses the graph of the function more than once, the function is not one-to-one.

For example, the function $x^3$ is one-to-one because its graph passes this test. Each horizontal line will intersect the curve at most once.
In contrast, a function like $x^2$ on the interval $(-\infty, \infty)$ is not one-to-one because it fails the horizontal line test.

Understanding one-to-one functions is crucial when dealing with function restrictions to ensure each input maps uniquely to the output.

Graph Sketching

Graph sketching is the process of visually representing a function to understand its behavior over different intervals. This involves plotting points and drawing curves based on the function's equation.

Begin by identifying intervals such as $(-\infty, -2)$ and $ (0, \infty)$, based on the variable conditions.
Next, sketch simpler functions like $x^3$ for $(-\infty, -2)$. Recognize that $x^3$ is an increasing curve, smoothly passing through negative x-values.
For $x^2$ on $ (0, \infty)$, recognize how it increases steeply from the origin, making a parabola upwards.

Connecting the intervals with a line on $[-2, 0]$ helps to illustrate parts where the graph is not one-to-one. Sketching this cleanly on paper ensures clarity in visual learning and interpretation.

Function Intervals

Function intervals represent specific sections of a function's domain where particular rules or behaviors apply. In piecewise functions, these intervals help determine which function expression to use.

An interval like $(-\infty, -2)$ implies examining behavior to the left of a point on the x-axis, using $x^3$ in this context.
Simultaneously, the interval $(0, \infty)$ focuses on $x^2$ to describe behavior to the right of the origin.
Bridging two intervals with a line segment such as $[-2, 0]$ highlights the joining of function segments, allowing discontinuities or different expressions within a comprehensive graph structure.

Understanding intervals is essential for interpreting complex functions. It allows each section of the function to be analyzed independently, aiding comprehension and problem-solving.

91影视

Sketch a graph of a function that is one-to-one on the intervals \((-\infty,-2),\) and \((0, \infty)\) but is not one-to-one on \((-\infty, \infty)\).

Short Answer

Step by step solution

Define the piecewise function

Connect the two intervals

Sketch the graph

Key Concepts

One-to-One Functions

Graph Sketching

Function Intervals

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