QuestionGraph the piecewise function $f(x)=\left\{\begin{aligned} x, & x<\frac{1}{2} \\ 3 x-1, & x \geq \frac{1}{2} .\end{aligned}\right.$ and find its domain and range.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as a piecewise function, with different expressions for $x$ depending on whether $x$ is less than $\frac{1}{}$ or greater than or equal to $\frac{1}{}$ . . We need to find the domain and range of this function.

STEP 2

The domain of a function is the set of all possible input values (typically the "x" variable), which produce a valid output from a particular function. For this function, we can see that there are no restrictions on $x$ in either of the two expressions. Therefore, the domain is all real numbers.
$Domain \mathbb{R}$

STEP 3

The range of a function is the set of all possible output values (typically the "y" variable), which come from the function. To find the range, we need to consider the output of the function for all possible input values in the domain.

STEP 4

First, consider the output when $x<\frac{1}{2}$ . In this case, the function is defined as $f(x)=x$ . Since $x$ can be any real number less than $\frac{1}{2}$ , the output can also be any real number less than $\frac{1}{2}$ .

STEP 5

Next, consider the output when $x\geq\frac{1}{2}$ . In this case, the function is defined as $f(x)=3x-1$ . Since $x$ can be any real number greater than or equal to $\frac{1}{2}$ , the output can be any real number greater than or equal to $3*\frac{1}{2}-1=\frac{1}{2}$ .

STEP 6

Combining the outputs from both cases, we can see that the range of the function is all real numbers greater than or equal to $\frac{1}{2}$ .
$Range [ \frac{1}{2}, \infty)$

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QuestionGraph the piecewise function $f(x)=\left\{\begin{aligned} x, & x<\frac{1}{2} \\ 3 x-1, & x \geq \frac{1}{2} .\end{aligned}\right.$ and find its domain and range.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as a piecewise function, with different expressions for $x$ depending on whether $x$ is less than $\frac{1}{}$ or greater than or equal to $\frac{1}{}$ . . We need to find the domain and range of this function.

STEP 2

STEP 3

The range of a function is the set of all possible output values (typically the "y" variable), which come from the function. To find the range, we need to consider the output of the function for all possible input values in the domain.

STEP 4

First, consider the output when $x<\frac{1}{2}$ . In this case, the function is defined as $f(x)=x$ . Since $x$ can be any real number less than $\frac{1}{2}$ , the output can also be any real number less than $\frac{1}{2}$ .

STEP 5

Next, consider the output when $x\geq\frac{1}{2}$ . In this case, the function is defined as $f(x)=3x-1$ . Since $x$ can be any real number greater than or equal to $\frac{1}{2}$ , the output can be any real number greater than or equal to $3*\frac{1}{2}-1=\frac{1}{2}$ .

STEP 6

Combining the outputs from both cases, we can see that the range of the function is all real numbers greater than or equal to $\frac{1}{2}$ .
$Range [ \frac{1}{2}, \infty)$

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QuestionGraph the piecewise function f(x)={x,x<123x−1,x≥12.f(x)=\left\{\begin{aligned} x, & x<\frac{1}{2} \\ 3 x-1, & x \geq \frac{1}{2} .\end{aligned}\right.f(x)=⎩⎨⎧​x,3x−1,​x<21​x≥21​.​ and find its domain and range.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x)f(x) is defined as a piecewise function, with different expressions for xxx depending on whether xxx is less than 1\frac{1}{}1​ or greater than or equal to 1\frac{1}{}1​. . We need to find the domain and range of this function.

STEP 2

STEP 3

The range of a function is the set of all possible output values (typically the "y" variable), which come from the function. To find the range, we need to consider the output of the function for all possible input values in the domain.

STEP 4

First, consider the output when x<12x<\frac{1}{2}x<21​. In this case, the function is defined as f(x)=xf(x)=xf(x)=x. Since xxx can be any real number less than 12\frac{1}{2}21​, the output can also be any real number less than 12\frac{1}{2}21​.

STEP 5

STEP 6

Combining the outputs from both cases, we can see that the range of the function is all real numbers greater than or equal to 12\frac{1}{2}21​.Range[12,∞)Range [ \frac{1}{2}, \infty)Range[21​,∞)

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QuestionGraph the piecewise function $f(x)=\left\{\begin{aligned} x, & x<\frac{1}{2} \\ 3 x-1, & x \geq \frac{1}{2} .\end{aligned}\right.$ and find its domain and range.

Assumptions1. The function $f(x)$ is defined as a piecewise function, with different expressions for $x$ depending on whether $x$ is less than $\frac{1}{}$ or greater than or equal to $\frac{1}{}$ . . We need to find the domain and range of this function.

First, consider the output when $x<\frac{1}{2}$ . In this case, the function is defined as $f(x)=x$ . Since $x$ can be any real number less than $\frac{1}{2}$ , the output can also be any real number less than $\frac{1}{2}$ .

Combining the outputs from both cases, we can see that the range of the function is all real numbers greater than or equal to $\frac{1}{2}$ .
$Range [ \frac{1}{2}, \infty)$