QuestionGraph the function $f(x)=\left\{\begin{array}{ccc}\frac{1}{2} x & \text { if } & x \leq 0 \\ 3 & \text { if } & x>0\end{array}\right.$ and find its range.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is a piecewise function defined as $\frac{1}{} x$ for $x \leq0$ and $3$ for $x >0$ . . We need to graph this function and determine its range.

STEP 2

Let's start by graphing the first piece of the function, $\frac{1}{2} x$ for $x \leq0$ . This is a linear function with a slope of $\frac{1}{2}$ and it is defined for all $x$ less than or equal to $0$ .

STEP 3

We can start by plotting a few points. When $x =0$ , $f(x) = \frac{1}{2} \times0 =0$ . When $x = -1$ , $f(x) = \frac{1}{2} \times -1 = -\frac{1}{2}$ . When $x = -2$ , $f(x) = \frac{1}{2} \times -2 = -1$ .

STEP 4

Now, let's graph the second piece of the function, $3$ for $x >0$ . This is a constant function and it is defined for all $x$ greater than $0$ .

STEP 5

We can plot a few points for this piece as well. When $x =1$ , $f(x) =3$ . When $x =2$ , $f(x) =3$ .

STEP 6

Now, let's put these two pieces together to graph the entire function. For $x \leq0$ , we graph the line $\frac{1}{2} x$ and for $x >0$ , we graph the horizontal line $y =3$ .

STEP 7

The graph of the function $f(x)$ will look like a line with slope $\frac{1}{2}$ for $x \leq0$ and a horizontal line at $y =3$ for $x >0$ .

STEP 8

Now, let's determine the range of the function. The range of a function is the set of all possible output values (y-values).

STEP 9

For the piece $\frac{}{2} x$ for $x \leq$ , the range is all y-values less than or equal to $$.

STEP 10

For the piece $3$ for $x >0$ , the range is simply $3$ .

STEP 11

Putting these two pieces together, the range of the function $f(x)$ is all y-values less than or equal to $0$ and $3$ .
So, the range of the function is $(-\infty,0] \cup \{3\}$ .

Was this helpful?

QuestionGraph the function $f(x)=\left\{\begin{array}{ccc}\frac{1}{2} x & \text { if } & x \leq 0 \\ 3 & \text { if } & x>0\end{array}\right.$ and find its range.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is a piecewise function defined as $\frac{1}{} x$ for $x \leq0$ and $3$ for $x >0$ . . We need to graph this function and determine its range.

STEP 2

Let's start by graphing the first piece of the function, $\frac{1}{2} x$ for $x \leq0$ . This is a linear function with a slope of $\frac{1}{2}$ and it is defined for all $x$ less than or equal to $0$ .

STEP 3

We can start by plotting a few points. When $x =0$ , $f(x) = \frac{1}{2} \times0 =0$ . When $x = -1$ , $f(x) = \frac{1}{2} \times -1 = -\frac{1}{2}$ . When $x = -2$ , $f(x) = \frac{1}{2} \times -2 = -1$ .

STEP 4

Now, let's graph the second piece of the function, $3$ for $x >0$ . This is a constant function and it is defined for all $x$ greater than $0$ .

STEP 5

We can plot a few points for this piece as well. When $x =1$ , $f(x) =3$ . When $x =2$ , $f(x) =3$ .

STEP 6

Now, let's put these two pieces together to graph the entire function. For $x \leq0$ , we graph the line $\frac{1}{2} x$ and for $x >0$ , we graph the horizontal line $y =3$ .

STEP 7

The graph of the function $f(x)$ will look like a line with slope $\frac{1}{2}$ for $x \leq0$ and a horizontal line at $y =3$ for $x >0$ .

STEP 8

Now, let's determine the range of the function. The range of a function is the set of all possible output values (y-values).

STEP 9

For the piece $\frac{}{2} x$ for $x \leq$ , the range is all y-values less than or equal to $$.

STEP 10

For the piece $3$ for $x >0$ , the range is simply $3$ .

STEP 11

Putting these two pieces together, the range of the function $f(x)$ is all y-values less than or equal to $0$ and $3$ .
So, the range of the function is $(-\infty,0] \cup \{3\}$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionGraph the function f(x)={12x if x≤03 if x>0f(x)=\left\{\begin{array}{ccc}\frac{1}{2} x & \text { if } & x \leq 0 \\ 3 & \text { if } & x>0\end{array}\right.f(x)={21​x3​ if if ​x≤0x>0​ and find its range.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x)f(x) is a piecewise function defined as 1x\frac{1}{} x1​x for x≤0x \leq0x≤0 and 333 for x>0x >0x>0. . We need to graph this function and determine its range.

STEP 2

Let's start by graphing the first piece of the function, 12x\frac{1}{2} x21​x for x≤0x \leq0x≤0. This is a linear function with a slope of 12\frac{1}{2}21​ and it is defined for all xxx less than or equal to 000.

STEP 3

STEP 4

Now, let's graph the second piece of the function, 333 for x>0x >0x>0. This is a constant function and it is defined for all xxx greater than 000.

STEP 5

We can plot a few points for this piece as well. When x=1x =1x=1, f(x)=3f(x) =3f(x)=3. When x=2x =2x=2, f(x)=3f(x) =3f(x)=3.

STEP 6

Now, let's put these two pieces together to graph the entire function. For x≤0x \leq0x≤0, we graph the line 12x\frac{1}{2} x21​x and for x>0x >0x>0, we graph the horizontal line y=3y =3y=3.

STEP 7

The graph of the function f(x)f(x)f(x) will look like a line with slope 12\frac{1}{2}21​ for x≤0x \leq0x≤0 and a horizontal line at y=3y =3y=3 for x>0x >0x>0.

STEP 8

Now, let's determine the range of the function. The range of a function is the set of all possible output values (y-values).

STEP 9

For the piece 2x\frac{}{2} x2​x for x≤x \leqx≤, the range is all y-values less than or equal to $$.

STEP 10

For the piece 333 for x>0x >0x>0, the range is simply 333.

STEP 11

Putting these two pieces together, the range of the function f(x)f(x)f(x) is all y-values less than or equal to 000 and 333.So, the range of the function is (−∞,0]∪{3}(-\infty,0] \cup \{3\}(−∞,0]∪{3}.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionGraph the function $f(x)=\left\{\begin{array}{ccc}\frac{1}{2} x & \text { if } & x \leq 0 \\ 3 & \text { if } & x>0\end{array}\right.$ and find its range.

Assumptions1. The function $f(x)$ is a piecewise function defined as $\frac{1}{} x$ for $x \leq0$ and $3$ for $x >0$ . . We need to graph this function and determine its range.

Let's start by graphing the first piece of the function, $\frac{1}{2} x$ for $x \leq0$ . This is a linear function with a slope of $\frac{1}{2}$ and it is defined for all $x$ less than or equal to $0$ .

Now, let's graph the second piece of the function, $3$ for $x >0$ . This is a constant function and it is defined for all $x$ greater than $0$ .

We can plot a few points for this piece as well. When $x =1$ , $f(x) =3$ . When $x =2$ , $f(x) =3$ .

Now, let's put these two pieces together to graph the entire function. For $x \leq0$ , we graph the line $\frac{1}{2} x$ and for $x >0$ , we graph the horizontal line $y =3$ .

The graph of the function $f(x)$ will look like a line with slope $\frac{1}{2}$ for $x \leq0$ and a horizontal line at $y =3$ for $x >0$ .

For the piece $\frac{}{2} x$ for $x \leq$ , the range is all y-values less than or equal to $$.

For the piece $3$ for $x >0$ , the range is simply $3$ .

Putting these two pieces together, the range of the function $f(x)$ is all y-values less than or equal to $0$ and $3$ .
So, the range of the function is $(-\infty,0] \cup \{3\}$ .