Problem 68 The domain of each piecewise fun... [FREE SOLUTION]

Chapter 1: Problem 68

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

Short Answer

Expert verified

The range of the function $f(x)$ is $[-1/2, \infty)$

Step by step solution

- Understand the Piecewise Function

The given function $f(x)$ is a piecewise function. A piecewise function is a function that is defined by several different formulas or expressions, depending on what range of input, or x-values, it鈥檚 applied to. In this case, $f(x)$ is defined by two different formulas: $-\frac{1}{2} x^{2}$ when $x<1$ and $2x+1$ when $x \geq 1$.

- Graph the First Piece of Function

Begin by graphing the first piece of the function: $-\frac{1}{2} x^{2}$. This equation represents a parabola. However, according to the function rule, this only applies for $x<1$, so one should stop the graph at x=1. Remember, at x=1, one needs to leave a open circle because that point is not included in this piece.

- Graph the Second Piece of Function

Now, graph the second piece of the function: $2x+1$. This equation represents a straight line. Described by the function rule, this applies for $x \geq 1$. Start this piece at x=1 and make sure to represent x=1 with a filled circle because the point is included in this piece according to the function definition.

- Find the Range

With the graph completed, to determine the range one can inspect the y-values taken by the function. The rule for determining is as follows: The lowest y-value to the highest y-value seen on the graph is the range of the function. The graph shows that the smallest value y can take is -1/2 and it increases without limit as x also increases so the range is $[-1/2, \infty)$

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Key Concepts

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

Understanding Domain and Range

The domain of a function refers to the set of all possible input values (x-values) that the function can handle. For the piecewise function presented in the exercise, the domain is

All real numbers, which can be expressed as $(-\infty, \infty)$.

This is because both segments of the piecewise function, the quadratic piece for

$ x < 1 $

and the linear piece for

$ x \geq 1 $,

are defined for all real numbers.The range, on the other hand, signifies all the possible output values (y-values) the function can yield. From the graph of our function, it starts at

$-1/2 $ for the quadratic section
It rises indefinitely due to the linear section, meaning the range is

$[-1/2, \infty)$. It鈥檚 vital to know both domain and range as they give us complete knowledge about what the function can accept and produce.

Graphing Functions

Graphing a piecewise function can be a little tricky, but with practice, it becomes more manageable. Let's break this down:- **Step 1:** Draw the graph of the quadratic function $-\frac{1}{2} x^{2}$ for

$ x < 1 $. Since this is a parabola, it'll open downward due to the negative coefficient on the $ x^{2} $ term.
Mark an open circle at $ x = 1 $ because it's not included in this part of the function.

- **Step 2:** Now, graph the linear function $2x+1$ for

$ x \geq 1 $. Start it from
$ (1, 3) $ since at
$ x = 1 $,

$ f(x) = 3 $.
- Connect the line upward, and put a filled circle at x=1, representing that the point is included.Graphing in segments while adhering to each rule is key to accurately represent a piecewise function on a graph.

Quadratic Functions

Quadratic functions are fundamental in mathematics, often displayed in the standard form:

$ ax^{2} + bx + c $.

A distinctive feature to recognize is their parabolic shape, which can either be upward or downward. In our case, $-\frac{1}{2} x^{2}$

is downward opening.

This is because the coefficient

in front of $ x^{2} $ is negative.

Here, - **Vertex:** Refers to the point at which the parabola changes direction. As

$ f(x) = -\frac{1}{2} x^{2}$,
vertex is at (0, 0)

- **Axis of Symmetry:** A vertical line passing through the vertex. For the quadratic section, it's

$ x = 0 $.

Quadratic functions pop up often in real-world scenarios involving projectile motion and areas where modeling and real-life applications meet.

Linear Functions

Linear functions are simpler and defined by the formula:

$ y = mx + b $.

Where

$ m $ represents the slope
$ b $ is the y-intercept.

In our piecewise function,

$2x + 1$

for

$ x \geq 1 $ defines the linear part.

- **Slope:** The rate of change or steepness dictates that for every increase in x by 1 unit, y increases by 2 units.

The line slopes upwards.

- **Y-Intercept:** The point where the line crosses the y-axis is at

(0, 1).

In mathematics, understanding linear functions allows us to predict outcomes and trends, making them crucial in fields such as business and science where constant rates of change are encountered.

91影视

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

Short Answer

Step by step solution

- Understand the Piecewise Function

- Graph the First Piece of Function

- Graph the Second Piece of Function

- Find the Range

Key Concepts

Understanding Domain and Range

Graphing Functions

Quadratic Functions

Linear Functions

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