Problem 62

Question

The domain of each piecewise function $i s(-\infty, \infty)$ a. Graph each function. b. Use your graph to determine the function's range. $$f(x)=\left\\{\begin{array}{lll} \frac{1}{2} x & \text { if } & x \leq 0 \\ 3 & \text { if } & x>0 \end{array}\right.$$

Step-by-Step Solution

Verified

Answer

The graph of the function consists of a straight line from $-\infty$ to $x=0$ and a horizontal line at $y=3$ for $x>0$. The range of the piecewise function is $-\infty \leq f(x) \leq 0$ and $f(x) = 3$.

1Step 1: Resolution of the function for $x \leq 0$

For the range $x \leq 0$, the function can be written as $f(x) = 1/2x$. This is a straight line with a slope of $1/2$. As $x$ ranges from $-\infty$ to $0$, $f(x)$ ranges from $-\infty$ to $0$.

2Step 2: Resolution of the function for $x > 0$

For the range $x > 0$, the function can be written as $f(x) = 3$. This is a horizontal line, at $y=3$, extending from $x=0$ to $\infty$. So, for $x$ ranging from just greater than $0$ to $\infty$, $f(x)=3$.

3Step 3: Combine the component functions

To combine both component functions, recall that the given function has its entire domain at $-\infty < x < \infty$. Therefore, the overall output or range of the function includes both parts, which are $-\infty < f(x) \leq 0$ and $f(x) = 3$.

4Step 4: Sketch the Graph

To graph the function, plot the straight line for $x \leq 0$, which begins from $-\infty$ and ends on y-axis at $y=0$. Then, draw a line parallel to x-axis at $y=3$ for $x > 0$. An open circle should be used to indicate $x>0$ because $x>0$ are not included in the second component function.

Key Concepts

Domain and Range in Piecewise FunctionsGraphing FunctionsUnderstanding Linear Functions

Domain and Range in Piecewise Functions

Understanding the domain and range of a piecewise function is crucial. The **domain** refers to all possible input values (x-values) that the function can handle. For the given piecewise function, the domain is all real numbers, expressed as i.e., $-\infty < x < \infty$.
This means that any real number can be plugged into the function.The **range** is the set of all possible output values (y-values). Since the function has two parts, the range needs to consider each. For $x \leq 0$, the function $f(x) = \frac{1}{2}x$ potentially outputs values from $-\infty$ to $0$.

The graph shows these outputs stretching up to and including $0$.
For $x > 0$, the output is always $3$.

Therefore, the complete range for this piecewise function includes all the outputs from $-\infty$ to $0$ and also $y=3$. It requires considering both segments of the function.

Graphing Functions

Graphing piecewise functions involves plotting each segment based on its given rule and condition. For this function, we have two distinct parts to graph.

**Part 1:** $f(x) = \frac{1}{2}x$ for $x \leq 0$ represents a line with a slope of $\frac{1}{2}$.
To graph this part, start from any point where $x$ is less than or equal to $0$. For instance, at $x = 0$, $f(x)$ is $0$. The line continues downward as $x$ decreases.
**Part 2:** This segment is $f(x) = 3$ for all $x > 0$.
It is a horizontal line extending parallel to the $x$-axis, indicating the same value for any $x$ greater than $0$. The importance of using an open circle on the graph at $x=0$ indicates $3$ is not included when $x=0$, but rather starts just above.

Graphing each segment carefully ensures an accurate visual representation of the function.

Understanding Linear Functions

Linear functions are foundational in graphing and representing relationships in mathematics. They are characterized by constant rates of change, visualized as straight lines on a graph.In the given piecewise function, the first segment $f(x) = \frac{1}{2}x$ for $x \leq 0$ encapsulates a linear function.

**Constant Slope:** The slope here is $\frac{1}{2}$, depicting that for every unit increase in $x$, $f(x)$ increases by half a unit. This steady rate is typical of linear functions.
**Intercept:** It intersects the origin $(0,0)$ since any line through this point without vertical displacement is linear.

The second part of the function, $f(x)=3$ when $x>0$, also reflects a linear function despite being horizontal.
This indicates zero changes in $f(x)$ regardless of $x$, emphasizing the versatility of linearity for constant functions.

Problem 62

Other exercises in this chapter

Problem 62

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+x+y-\frac{1}{2

View solution

Problem 62

Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.

View solution

Problem 62

Find. a. $(f \circ g)(x)$ b. $(g \circ f)(x)$ c. $(f \circ g)(2)$ d. $(g \circ f)(2)$ $$f(x)=\sqrt{x}, g(x)=x+2$$

View solution

Problem 62

Begin by graphing the standard quadratic function, $f(x)=x^{2} .$ Then use transformations of this graph to graph the given function. $$g(x)=\frac{1}{2}(x-1)^

View solution