QuestionDefine the function $f(x)$ as: $f(x)=x$ if $x<0$ and $f(x)=x^{2}$ if $x \geq 0$ . Find its domain, intercepts, graph, and range.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined piecewise, with different rules for $x<0$ and $x \geq0$ . . We need to find the domain, intercepts, graph, and range of the function.

STEP 2

(a) To find the domain of the function, we need to identify the set of all possible $x$ values that the function can take. Since the function is defined for all real numbers, the domain of the function is all real numbers.
$Domain = (-\infty, +\infty)$

STEP 3

(b) To find the intercepts of the function, we need to find the $x$ and $y$ values where the function intersects the $x$ -axis and $y$ -axis.For the $x$ -intercept, we set $f(x) =0$ and solve for $x$ .For the $y$ -intercept, we set $x =0$ and solve for $f(x)$ .

STEP 4

First, let's find the $x$ -intercept.For $x<0$ , $f(x) = x$ . Setting this equal to zero gives $x =0$ .
For $x \geq0$ , $f(x) = x^2$ . Setting this equal to zero gives $x =0$ .
So, the $x$ -intercept is at $x =0$ .

STEP 5

Next, let's find the $y$ -intercept.Setting $x =0$ in the function gives $f(0) =0^2 =0$ .
So, the $y$ -intercept is at $y =0$ .

STEP 6

(c) To graph the function, we plot the function for $x<0$ and $x \geq0$ separately.
For $x<0$ , the function is a straight line passing through the origin with a slope of1.
For $x \geq0$ , the function is a parabola opening upwards with the vertex at the origin.

STEP 7

(d) To find the range of the function based on the graph, we identify the set of all possible $y$ values that the function can take.
Looking at the graph, we see that the function takes all $y$ values greater than or equal to0.
So, the range of the function is $[0, +\infty)$ .

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QuestionDefine the function $f(x)$ as: $f(x)=x$ if $x<0$ and $f(x)=x^{2}$ if $x \geq 0$ . Find its domain, intercepts, graph, and range.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined piecewise, with different rules for $x<0$ and $x \geq0$ . . We need to find the domain, intercepts, graph, and range of the function.

STEP 2

(a) To find the domain of the function, we need to identify the set of all possible $x$ values that the function can take. Since the function is defined for all real numbers, the domain of the function is all real numbers.
$Domain = (-\infty, +\infty)$

STEP 3

(b) To find the intercepts of the function, we need to find the $x$ and $y$ values where the function intersects the $x$ -axis and $y$ -axis.For the $x$ -intercept, we set $f(x) =0$ and solve for $x$ .For the $y$ -intercept, we set $x =0$ and solve for $f(x)$ .

STEP 4

First, let's find the $x$ -intercept.For $x<0$ , $f(x) = x$ . Setting this equal to zero gives $x =0$ .
For $x \geq0$ , $f(x) = x^2$ . Setting this equal to zero gives $x =0$ .
So, the $x$ -intercept is at $x =0$ .

STEP 5

Next, let's find the $y$ -intercept.Setting $x =0$ in the function gives $f(0) =0^2 =0$ .
So, the $y$ -intercept is at $y =0$ .

STEP 6

(c) To graph the function, we plot the function for $x<0$ and $x \geq0$ separately.
For $x<0$ , the function is a straight line passing through the origin with a slope of1.
For $x \geq0$ , the function is a parabola opening upwards with the vertex at the origin.

STEP 7

(d) To find the range of the function based on the graph, we identify the set of all possible $y$ values that the function can take.
Looking at the graph, we see that the function takes all $y$ values greater than or equal to0.
So, the range of the function is $[0, +\infty)$ .

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QuestionDefine the function f(x)f(x)f(x) as: f(x)=xf(x)=xf(x)=x if x<0x<0x<0 and f(x)=x2f(x)=x^{2}f(x)=x2 if x≥0x \geq 0x≥0. Find its domain, intercepts, graph, and range.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x)f(x) is defined piecewise, with different rules for x<0x<0x<0 and x≥0x \geq0x≥0. . We need to find the domain, intercepts, graph, and range of the function.

STEP 2

(a) To find the domain of the function, we need to identify the set of all possible xxx values that the function can take. Since the function is defined for all real numbers, the domain of the function is all real numbers.Domain=(−∞,+∞)Domain = (-\infty, +\infty)Domain=(−∞,+∞)

STEP 3

(b) To find the intercepts of the function, we need to find the xxx and yyy values where the function intersects the xxx-axis and yyy-axis.For the xxx-intercept, we set f(x)=0f(x) =0f(x)=0 and solve for xxx.For the yyy-intercept, we set x=0x =0x=0 and solve for f(x)f(x)f(x).

STEP 4

First, let's find the xxx-intercept.For x<0x<0x<0, f(x)=xf(x) = xf(x)=x. Setting this equal to zero gives x=0x =0x=0.For x≥0x \geq0x≥0, f(x)=x2f(x) = x^2f(x)=x2. Setting this equal to zero gives x=0x =0x=0.So, the xxx-intercept is at x=0x =0x=0.

STEP 5

Next, let's find the yyy-intercept.Setting x=0x =0x=0 in the function gives f(0)=02=0f(0) =0^2 =0f(0)=02=0.So, the yyy-intercept is at y=0y =0y=0.

STEP 6

(c) To graph the function, we plot the function for x<0x<0x<0 and x≥0x \geq0x≥0 separately.For x<0x<0x<0, the function is a straight line passing through the origin with a slope of1.For x≥0x \geq0x≥0, the function is a parabola opening upwards with the vertex at the origin.

STEP 7

(d) To find the range of the function based on the graph, we identify the set of all possible yyy values that the function can take.Looking at the graph, we see that the function takes all yyy values greater than or equal to0.So, the range of the function is [0,+∞)[0, +\infty)[0,+∞).

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QuestionDefine the function $f(x)$ as: $f(x)=x$ if $x<0$ and $f(x)=x^{2}$ if $x \geq 0$ . Find its domain, intercepts, graph, and range.

Assumptions1. The function $f(x)$ is defined piecewise, with different rules for $x<0$ and $x \geq0$ . . We need to find the domain, intercepts, graph, and range of the function.

(a) To find the domain of the function, we need to identify the set of all possible $x$ values that the function can take. Since the function is defined for all real numbers, the domain of the function is all real numbers.
$Domain = (-\infty, +\infty)$

(b) To find the intercepts of the function, we need to find the $x$ and $y$ values where the function intersects the $x$ -axis and $y$ -axis.For the $x$ -intercept, we set $f(x) =0$ and solve for $x$ .For the $y$ -intercept, we set $x =0$ and solve for $f(x)$ .

First, let's find the $x$ -intercept.For $x<0$ , $f(x) = x$ . Setting this equal to zero gives $x =0$ .
For $x \geq0$ , $f(x) = x^2$ . Setting this equal to zero gives $x =0$ .
So, the $x$ -intercept is at $x =0$ .

Next, let's find the $y$ -intercept.Setting $x =0$ in the function gives $f(0) =0^2 =0$ .
So, the $y$ -intercept is at $y =0$ .

(c) To graph the function, we plot the function for $x<0$ and $x \geq0$ separately.
For $x<0$ , the function is a straight line passing through the origin with a slope of1.
For $x \geq0$ , the function is a parabola opening upwards with the vertex at the origin.

(d) To find the range of the function based on the graph, we identify the set of all possible $y$ values that the function can take.
Looking at the graph, we see that the function takes all $y$ values greater than or equal to0.
So, the range of the function is $[0, +\infty)$ .