QuestionDefine the function $f(x)=\begin{cases} 2+x & \text{if } x<0 \\ x^{2} & \text{if } x \geq 0 \end{cases}$ . Find its domain, intercepts, graph, and range.

Studdy Solution

STEP 1

Assumptions1. The function $f$ is defined as a piecewise function. . For $x<0$ , $f(x)=+x$ .
3. For $x \geq0$ , $f(x)=x^{}$ .

STEP 2

(a) To find the domain of the function, we need to determine the set of all possible values of $x$ that make the function defined.The domain of $f(x)=2+x$ is all real numbers, since $x$ can take any real number value.The domain of $f(x)=x^{2}$ is also all real numbers, since $x$ can take any real number value.Therefore, the domain of the function $f$ is all real numbers.

STEP 3

(b) To find the intercepts of the function, we need to find the $x$ -intercepts and the $y$ -intercept.
The $x$ -intercepts are the values of $x$ for which $f(x)=0$ .The $y$ -intercept is the value of $f(x)$ when $x=0$ .

STEP 4

First, let's find the $x$ -intercepts.
For $x<0$ , set $f(x)=0$ and solve for $x$ .
$0=2+x$

STEP 5

olve the equation for $x$ .
$x=0-2=-2$ So, $x=-2$ is an $x$ -intercept for the function when $x<0$ .

STEP 6

For $x \geq0$ , set $f(x)=0$ and solve for $x$ .
$0=x^{2}$

STEP 7

olve the equation for $x$ .
$x=\sqrt{0}=0$ So, $x=0$ is an $x$ -intercept for the function when $x \geq0$ .

STEP 8

Now, let's find the $y$ -intercept.
The $y$ -intercept is the value of $f(x)$ when $x=0$ .
$f(0)=0^{2}=0$ So, the $y$ -intercept is $0$ .

STEP 9

(c) To graph the function, we need to plot the function for $x<$ and $x \geq$ separately.
For $x<$ , the function $f(x)=2+x$ is a straight line with a slope of $$ and a $y$-intercept of $2$.
For $x \geq$ , the function $f(x)=x^{2}$ is a parabola opening upwards with the vertex at the origin.

STEP 10

(d) To find the range of the function, we need to find the set of all possible values of $f(x)$ .
For $x<0$ , the function $f(x)=2+x$ can take any real number value less than $2$ .
For $x \geq0$ , the function $f(x)=x^{2}$ can take any real number value greater than or equal to $0$ .
Therefore, the range of the function $f$ is $(-\infty,2) \cup [0,+\infty)$ .

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QuestionDefine the function $f(x)=\begin{cases} 2+x & \text{if } x<0 \\ x^{2} & \text{if } x \geq 0 \end{cases}$ . Find its domain, intercepts, graph, and range.

Studdy Solution

STEP 1

Assumptions1. The function $f$ is defined as a piecewise function. . For $x<0$ , $f(x)=+x$ .
3. For $x \geq0$ , $f(x)=x^{}$ .

STEP 2

STEP 3

(b) To find the intercepts of the function, we need to find the $x$ -intercepts and the $y$ -intercept.
The $x$ -intercepts are the values of $x$ for which $f(x)=0$ .The $y$ -intercept is the value of $f(x)$ when $x=0$ .

STEP 4

First, let's find the $x$ -intercepts.
For $x<0$ , set $f(x)=0$ and solve for $x$ .
$0=2+x$

STEP 5

olve the equation for $x$ .
$x=0-2=-2$ So, $x=-2$ is an $x$ -intercept for the function when $x<0$ .

STEP 6

For $x \geq0$ , set $f(x)=0$ and solve for $x$ .
$0=x^{2}$

STEP 7

olve the equation for $x$ .
$x=\sqrt{0}=0$ So, $x=0$ is an $x$ -intercept for the function when $x \geq0$ .

STEP 8

Now, let's find the $y$ -intercept.
The $y$ -intercept is the value of $f(x)$ when $x=0$ .
$f(0)=0^{2}=0$ So, the $y$ -intercept is $0$ .

STEP 9

(c) To graph the function, we need to plot the function for $x<$ and $x \geq$ separately.
For $x<$ , the function $f(x)=2+x$ is a straight line with a slope of $$ and a $y$-intercept of $2$.
For $x \geq$ , the function $f(x)=x^{2}$ is a parabola opening upwards with the vertex at the origin.

STEP 10

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QuestionDefine the function f(x)={2+xif x<0x2if x≥0f(x)=\begin{cases} 2+x & \text{if } x<0 \\ x^{2} & \text{if } x \geq 0 \end{cases}f(x)={2+xx2​if x<0if x≥0​. Find its domain, intercepts, graph, and range.

Studdy Solution

STEP 1

Assumptions1. The function fff is defined as a piecewise function. . For x<0x<0x<0, f(x)=+xf(x)=+xf(x)=+x.3. For x≥0x \geq0x≥0, f(x)=xf(x)=x^{}f(x)=x.

STEP 2

STEP 3

(b) To find the intercepts of the function, we need to find the xxx-intercepts and the yyy-intercept.The xxx-intercepts are the values of xxx for which f(x)=0f(x)=0f(x)=0.The yyy-intercept is the value of f(x)f(x)f(x) when x=0x=0x=0.

STEP 4

First, let's find the xxx-intercepts.For x<0x<0x<0, set f(x)=0f(x)=0f(x)=0 and solve for xxx.0=2+x0=2+x0=2+x

STEP 5

olve the equation for xxx.x=0−2=−2x=0-2=-2x=0−2=−2So, x=−2x=-2x=−2 is an xxx-intercept for the function when x<0x<0x<0.

STEP 6

For x≥0x \geq0x≥0, set f(x)=0f(x)=0f(x)=0 and solve for xxx.0=x20=x^{2}0=x2

STEP 7

olve the equation for xxx.x=0=0x=\sqrt{0}=0x=0​=0So, x=0x=0x=0 is an xxx-intercept for the function when x≥0x \geq0x≥0.

STEP 8

Now, let's find the yyy-intercept.The yyy-intercept is the value of f(x)f(x)f(x) when x=0x=0x=0.f(0)=02=0f(0)=0^{2}=0f(0)=02=0So, the yyy-intercept is 000.

STEP 9

STEP 10

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QuestionDefine the function $f(x)=\begin{cases} 2+x & \text{if } x<0 \\ x^{2} & \text{if } x \geq 0 \end{cases}$ . Find its domain, intercepts, graph, and range.

Assumptions1. The function $f$ is defined as a piecewise function. . For $x<0$ , $f(x)=+x$ .
3. For $x \geq0$ , $f(x)=x^{}$ .

(b) To find the intercepts of the function, we need to find the $x$ -intercepts and the $y$ -intercept.
The $x$ -intercepts are the values of $x$ for which $f(x)=0$ .The $y$ -intercept is the value of $f(x)$ when $x=0$ .

First, let's find the $x$ -intercepts.
For $x<0$ , set $f(x)=0$ and solve for $x$ .
$0=2+x$

olve the equation for $x$ .
$x=0-2=-2$ So, $x=-2$ is an $x$ -intercept for the function when $x<0$ .

For $x \geq0$ , set $f(x)=0$ and solve for $x$ .
$0=x^{2}$

olve the equation for $x$ .
$x=\sqrt{0}=0$ So, $x=0$ is an $x$ -intercept for the function when $x \geq0$ .

Now, let's find the $y$ -intercept.
The $y$ -intercept is the value of $f(x)$ when $x=0$ .
$f(0)=0^{2}=0$ So, the $y$ -intercept is $0$ .