QuestionDefine the function $f$ : $f(x) = -3x + 4$ for $x < 1$ and $f(x) = 4x - 3$ for $x \geq 1$ . Find its domain, intercepts, graph, and range.

Studdy Solution

STEP 1

Assumptions1. The function $f$ is defined as a piecewise function with two parts - $-3x +4$ if $x <1$ - $4x -3$ if $x \geq1$ . We need to find the domain, intercepts, graph, and range of the function.

STEP 2

(a) 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 piecewise function, the domain is all real numbers because there are no restrictions on $x$ in either piece of the function.In interval notation, the domain is represented as $(-\infty, \infty)$

STEP 3

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

STEP 4

First, let's find the y-intercept by setting $x =0$ in the appropriate part of the piecewise function. Since $0 <1$ , we use the first part of the function $f(0) = -3(0) +4 =4$ So, the y-intercept is $4$ .

STEP 5

Next, let's find the x-intercepts by setting $f(x) =0$ in each part of the function and solving for $x$ .
In the first part of the function, $-3x +4 =0$ when $x <1$ $-3x +4 =0 \Rightarrow x =4/3$ However, $4/3$ is not less than $1$ , so there are no x-intercepts in this part of the function.

STEP 6

In the second part of the function, $4x -3 =0$ when $x \geq1$ $4x -3 =0 \Rightarrow x =3/4$ However, $3/4$ is not greater than or equal to $1$ , so there are no x-intercepts in this part of the function either.
Therefore, the function has no x-intercepts.

STEP 7

(c) To graph the function, we plot the y-intercept and the points where the function changes from one part to the other (i.e., the point $(1, f(1))$ ).Since $f(1) =4(1) -3 =1$ , the point $(1,1)$ is on the graph.
The graph is a line with slope $-3$ and y-intercept $4$ for $x <1$ , and a line with slope $4$ and y-intercept $-3$ for $x \geq1$ . The two lines intersect at the point $(1,1)$ .

STEP 8

(d) The range of a function is the set of all possible output values (typically the "y" variable). Looking at the graph, we can see that the function covers all real numbers.So, the range of the function is also all real numbers, or in interval notation $(-\infty, \infty)$

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QuestionDefine the function $f$ : $f(x) = -3x + 4$ for $x < 1$ and $f(x) = 4x - 3$ for $x \geq 1$ . Find its domain, intercepts, graph, and range.

Studdy Solution

STEP 1

Assumptions1. The function $f$ is defined as a piecewise function with two parts - $-3x +4$ if $x <1$ - $4x -3$ if $x \geq1$ . We need to find the domain, intercepts, graph, and range of the function.

STEP 2

STEP 3

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

STEP 4

First, let's find the y-intercept by setting $x =0$ in the appropriate part of the piecewise function. Since $0 <1$ , we use the first part of the function $f(0) = -3(0) +4 =4$ So, the y-intercept is $4$ .

STEP 5

Next, let's find the x-intercepts by setting $f(x) =0$ in each part of the function and solving for $x$ .
In the first part of the function, $-3x +4 =0$ when $x <1$ $-3x +4 =0 \Rightarrow x =4/3$ However, $4/3$ is not less than $1$ , so there are no x-intercepts in this part of the function.

STEP 6

In the second part of the function, $4x -3 =0$ when $x \geq1$ $4x -3 =0 \Rightarrow x =3/4$ However, $3/4$ is not greater than or equal to $1$ , so there are no x-intercepts in this part of the function either.
Therefore, the function has no x-intercepts.

STEP 7

STEP 8

(d) The range of a function is the set of all possible output values (typically the "y" variable). Looking at the graph, we can see that the function covers all real numbers.So, the range of the function is also all real numbers, or in interval notation $(-\infty, \infty)$

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QuestionDefine the function fff: f(x)=−3x+4f(x) = -3x + 4f(x)=−3x+4 for x<1x < 1x<1 and f(x)=4x−3f(x) = 4x - 3f(x)=4x−3 for x≥1x \geq 1x≥1. Find its domain, intercepts, graph, and range.

Studdy Solution

STEP 1

Assumptions1. The function fff is defined as a piecewise function with two parts - −3x+4-3x +4−3x+4 if x<1x <1x<1 - 4x−34x -34x−3 if x≥1x \geq1x≥1 . We need to find the domain, intercepts, graph, and range of the function.

STEP 2

STEP 3

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

STEP 4

First, let's find the y-intercept by setting x=0x =0x=0 in the appropriate part of the piecewise function. Since 0<10 <10<1, we use the first part of the functionf(0)=−3(0)+4=4f(0) = -3(0) +4 =4f(0)=−3(0)+4=4So, the y-intercept is 444.

STEP 5

STEP 6

STEP 7

STEP 8

(d) The range of a function is the set of all possible output values (typically the "y" variable). Looking at the graph, we can see that the function covers all real numbers.So, the range of the function is also all real numbers, or in interval notation(−∞,∞)(-\infty, \infty)(−∞,∞)

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

QuestionDefine the function $f$ : $f(x) = -3x + 4$ for $x < 1$ and $f(x) = 4x - 3$ for $x \geq 1$ . Find its domain, intercepts, graph, and range.

Assumptions1. The function $f$ is defined as a piecewise function with two parts - $-3x +4$ if $x <1$ - $4x -3$ if $x \geq1$ . We need to find the domain, intercepts, graph, and range of the function.

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

First, let's find the y-intercept by setting $x =0$ in the appropriate part of the piecewise function. Since $0 <1$ , we use the first part of the function $f(0) = -3(0) +4 =4$ So, the y-intercept is $4$ .

(d) The range of a function is the set of all possible output values (typically the "y" variable). Looking at the graph, we can see that the function covers all real numbers.So, the range of the function is also all real numbers, or in interval notation $(-\infty, \infty)$