QuestionFind the sum function $(f+g)(x)$ for $f(x)=5x+4$ if $x<2$ and $x^2+4x$ if $x \geq 2$ , and $g(x)=-3x+1$ if $x \leq 0$ and $g(x)=x-7$ if $x > 0$ .

Studdy Solution

STEP 1

Assumptions1. We have two functions, $f(x)$ and $g(x)$ , which are defined differently for different ranges of $x$ . . The function $f(x)$ is defined as $5x+4$ for $x \geq$ and as $x^+4x$ otherwise.
3. The function $g(x)$ is defined as $-3x+1$ for $x \leq0$ and as $x-7$ for $x >0$ .
4. We need to find the sum function $(f+g)(x)$ .

STEP 2

The sum function $(f+g)(x)$ is defined as $f(x) + g(x)$ . Since both $f(x)$ and $g(x)$ are defined differently for different ranges of $x$ , we need to consider these ranges separately.

STEP 3

Let's first consider the range $x \leq0$ . In this range, $f(x) = x^2+x$ and $g(x) = -3x+1$ . So, the sum function $(f+g)(x)$ is $(f+g)(x) = f(x) + g(x) = x^2+x + (-3x+1)$

STEP 4

implify the expression for $(f+g)(x)$ for $x \leq0$ .
$(f+g)(x) = x^2+4x -3x +1 = x^2+x+1$

STEP 5

Next, consider the range $0 < x <2$ . In this range, $f(x) = x^2+4x$ and $g(x) = x-7$ . So, the sum function $(f+g)(x)$ is $(f+g)(x) = f(x) + g(x) = x^2+4x + (x-7)$

STEP 6

implify the expression for $(f+g)(x)$ for $0 < x <2$ .
$(f+g)(x) = x^2+4x + x - = x^2+5x-$

STEP 7

Finally, consider the range $x \geq2$ . In this range, $f(x) =5x+4$ and $g(x) = x-7$ . So, the sum function $(f+g)(x)$ is $(f+g)(x) = f(x) + g(x) =5x+4 + (x-7)$

STEP 8

implify the expression for $(f+g)(x)$ for $x \geq2$ .
$(f+g)(x) =5x+4 + x -7 =6x-3$

STEP 9

Now we have the sum function $(f+g)(x)$ for all possible ranges of $x$ . So, we can write $(f+g)(x) = \left\{\begin{array}{c}x^2+x+ \text { if } x \leq \\ x^2+5x-7 \text { if } < x <2 \\6x-3 \text { if } x \geq2\end{array}\right.$
This is the sum function $(f+g)(x)$ .

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QuestionFind the sum function $(f+g)(x)$ for $f(x)=5x+4$ if $x<2$ and $x^2+4x$ if $x \geq 2$ , and $g(x)=-3x+1$ if $x \leq 0$ and $g(x)=x-7$ if $x > 0$ .

Studdy Solution

STEP 1

STEP 2

The sum function $(f+g)(x)$ is defined as $f(x) + g(x)$ . Since both $f(x)$ and $g(x)$ are defined differently for different ranges of $x$ , we need to consider these ranges separately.

STEP 3

Let's first consider the range $x \leq0$ . In this range, $f(x) = x^2+x$ and $g(x) = -3x+1$ . So, the sum function $(f+g)(x)$ is $(f+g)(x) = f(x) + g(x) = x^2+x + (-3x+1)$

STEP 4

implify the expression for $(f+g)(x)$ for $x \leq0$ .
$(f+g)(x) = x^2+4x -3x +1 = x^2+x+1$

STEP 5

Next, consider the range $0 < x <2$ . In this range, $f(x) = x^2+4x$ and $g(x) = x-7$ . So, the sum function $(f+g)(x)$ is $(f+g)(x) = f(x) + g(x) = x^2+4x + (x-7)$

STEP 6

implify the expression for $(f+g)(x)$ for $0 < x <2$ .
$(f+g)(x) = x^2+4x + x - = x^2+5x-$

STEP 7

Finally, consider the range $x \geq2$ . In this range, $f(x) =5x+4$ and $g(x) = x-7$ . So, the sum function $(f+g)(x)$ is $(f+g)(x) = f(x) + g(x) =5x+4 + (x-7)$

STEP 8

implify the expression for $(f+g)(x)$ for $x \geq2$ .
$(f+g)(x) =5x+4 + x -7 =6x-3$

STEP 9

Now we have the sum function $(f+g)(x)$ for all possible ranges of $x$ . So, we can write $(f+g)(x) = \left\{\begin{array}{c}x^2+x+ \text { if } x \leq \\ x^2+5x-7 \text { if } < x <2 \\6x-3 \text { if } x \geq2\end{array}\right.$
This is the sum function $(f+g)(x)$ .

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QuestionFind the sum function (f+g)(x)(f+g)(x)(f+g)(x) for f(x)=5x+4f(x)=5x+4f(x)=5x+4 if x<2x<2x<2 and x2+4xx^2+4xx2+4x if x≥2x \geq 2x≥2, and g(x)=−3x+1g(x)=-3x+1g(x)=−3x+1 if x≤0x \leq 0x≤0 and g(x)=x−7g(x)=x-7g(x)=x−7 if x>0x > 0x>0.

Studdy Solution

STEP 1

STEP 2

The sum function (f+g)(x)(f+g)(x)(f+g)(x) is defined as f(x)+g(x)f(x) + g(x)f(x)+g(x). Since both f(x)f(x)f(x) and g(x)g(x)g(x) are defined differently for different ranges of xxx, we need to consider these ranges separately.

STEP 3

Let's first consider the range x≤0x \leq0x≤0. In this range, f(x)=x2+xf(x) = x^2+xf(x)=x2+x and g(x)=−3x+1g(x) = -3x+1g(x)=−3x+1. So, the sum function (f+g)(x)(f+g)(x)(f+g)(x) is(f+g)(x)=f(x)+g(x)=x2+x+(−3x+1)(f+g)(x) = f(x) + g(x) = x^2+x + (-3x+1)(f+g)(x)=f(x)+g(x)=x2+x+(−3x+1)

STEP 4

implify the expression for (f+g)(x)(f+g)(x)(f+g)(x) for x≤0x \leq0x≤0.(f+g)(x)=x2+4x−3x+1=x2+x+1(f+g)(x) = x^2+4x -3x +1 = x^2+x+1(f+g)(x)=x2+4x−3x+1=x2+x+1

STEP 5

Next, consider the range 0<x<20 < x <20<x<2. In this range, f(x)=x2+4xf(x) = x^2+4xf(x)=x2+4x and g(x)=x−7g(x) = x-7g(x)=x−7. So, the sum function (f+g)(x)(f+g)(x)(f+g)(x) is(f+g)(x)=f(x)+g(x)=x2+4x+(x−7)(f+g)(x) = f(x) + g(x) = x^2+4x + (x-7)(f+g)(x)=f(x)+g(x)=x2+4x+(x−7)

STEP 6

implify the expression for (f+g)(x)(f+g)(x)(f+g)(x) for 0<x<20 < x <20<x<2.(f+g)(x)=x2+4x+x−=x2+5x−(f+g)(x) = x^2+4x + x - = x^2+5x-(f+g)(x)=x2+4x+x−=x2+5x−

STEP 7

Finally, consider the range x≥2x \geq2x≥2. In this range, f(x)=5x+4f(x) =5x+4f(x)=5x+4 and g(x)=x−7g(x) = x-7g(x)=x−7. So, the sum function (f+g)(x)(f+g)(x)(f+g)(x) is(f+g)(x)=f(x)+g(x)=5x+4+(x−7)(f+g)(x) = f(x) + g(x) =5x+4 + (x-7)(f+g)(x)=f(x)+g(x)=5x+4+(x−7)

STEP 8

implify the expression for (f+g)(x)(f+g)(x)(f+g)(x) for x≥2x \geq2x≥2.(f+g)(x)=5x+4+x−7=6x−3(f+g)(x) =5x+4 + x -7 =6x-3(f+g)(x)=5x+4+x−7=6x−3

STEP 9

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QuestionFind the sum function $(f+g)(x)$ for $f(x)=5x+4$ if $x<2$ and $x^2+4x$ if $x \geq 2$ , and $g(x)=-3x+1$ if $x \leq 0$ and $g(x)=x-7$ if $x > 0$ .

The sum function $(f+g)(x)$ is defined as $f(x) + g(x)$ . Since both $f(x)$ and $g(x)$ are defined differently for different ranges of $x$ , we need to consider these ranges separately.

Let's first consider the range $x \leq0$ . In this range, $f(x) = x^2+x$ and $g(x) = -3x+1$ . So, the sum function $(f+g)(x)$ is $(f+g)(x) = f(x) + g(x) = x^2+x + (-3x+1)$

implify the expression for $(f+g)(x)$ for $x \leq0$ .
$(f+g)(x) = x^2+4x -3x +1 = x^2+x+1$

Next, consider the range $0 < x <2$ . In this range, $f(x) = x^2+4x$ and $g(x) = x-7$ . So, the sum function $(f+g)(x)$ is $(f+g)(x) = f(x) + g(x) = x^2+4x + (x-7)$

implify the expression for $(f+g)(x)$ for $0 < x <2$ .
$(f+g)(x) = x^2+4x + x - = x^2+5x-$

Finally, consider the range $x \geq2$ . In this range, $f(x) =5x+4$ and $g(x) = x-7$ . So, the sum function $(f+g)(x)$ is $(f+g)(x) = f(x) + g(x) =5x+4 + (x-7)$

implify the expression for $(f+g)(x)$ for $x \geq2$ .
$(f+g)(x) =5x+4 + x -7 =6x-3$