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QuestionFind the sums, differences, products, and quotients of the following functions:
1. f(x)=3x+4,g(x)=2x−1f(x)=3x+4, g(x)=2x-1
2. f(x)=2x−5,g(x)=4x2f(x)=2x-5, g(x)=4x^2
3. f(x)=x−4,g(x)=xf(x)=x-4, g(x)=\sqrt{x}

Studdy Solution

STEP 1

Assumptions1. The functions are f(x)=3x+4f(x)=3x+4 and g(x)=x−1g(x)=x-1 for the first part, f(x)=x−5f(x)=x-5 and g(x)=4xg(x)=4x^{} for the second part, and f(x)=x−4f(x)=x-4 and g(x)=xg(x)=\sqrt{x} for the third part. . The operations to be performed are addition, subtraction, multiplication, and division of functions.

STEP 2

First, we will find (f+g)(x)(f+g)(x) for the first part. This is done by adding the two functions together.
(f+g)(x)=f(x)+g(x)(f+g)(x) = f(x) + g(x)

STEP 3

Plug in the functions f(x)f(x) and g(x)g(x) to calculate (f+g)(x)(f+g)(x).
(f+g)(x)=(3x+)+(2x−1)(f+g)(x) = (3x+) + (2x-1)

STEP 4

implify the expression to find (f+g)(x)(f+g)(x).
(f+g)(x)=x+3(f+g)(x) =x +3

STEP 5

Now, we will find (f−g)(x)(f-g)(x) for the first part. This is done by subtracting g(x)g(x) from f(x)f(x).
(f−g)(x)=f(x)−g(x)(f-g)(x) = f(x) - g(x)

STEP 6

Plug in the functions f(x)f(x) and g(x)g(x) to calculate (f−g)(x)(f-g)(x).
(f−g)(x)=(3x+4)−(2x−1)(f-g)(x) = (3x+4) - (2x-1)

STEP 7

implify the expression to find (f−g)(x)(f-g)(x).
(f−g)(x)=x+5(f-g)(x) = x +5

STEP 8

Next, we will find (fg)(x)(fg)(x) for the first part. This is done by multiplying f(x)f(x) and g(x)g(x).
(fg)(x)=f(x)⋅g(x)(fg)(x) = f(x) \cdot g(x)

STEP 9

Plug in the functions f(x)f(x) and g(x)g(x) to calculate (fg)(x)(fg)(x).
(fg)(x)=(3x+4)⋅(2x−)(fg)(x) = (3x+4) \cdot (2x-)

STEP 10

Expand the expression to find (fg)(x)(fg)(x).
(fg)(x)=6x2−3x+8x−4=6x2+5x−4(fg)(x) =6x^{2} -3x +8x -4 =6x^{2} +5x -4

STEP 11

Lastly, we will find (fg)(x)\left(\frac{f}{g}\right)(x) for the first part. This is done by dividing f(x)f(x) by g(x)g(x).
(fg)(x)=f(x)g(x)\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}

STEP 12

Plug in the functions f(x)f(x) and g(x)g(x) to calculate (fg)(x)\left(\frac{f}{g}\right)(x).
(fg)(x)=x+42x−\left(\frac{f}{g}\right)(x) = \frac{x+4}{2x-}

STEP 13

We will repeat the steps2-12 for the second and third parts with the respective functions.
For the second part, we have(f+g)(x)=2x−5+x2=x2+2x−5(f+g)(x) =2x -5 +x^{2} =x^{2} +2x -5(f−g)(x)=2x−5−x2=−x2+2x−5(f-g)(x) =2x -5 -x^{2} = -x^{2} +2x -5(fg)(x) = (2x -5) \cdotx^{2} =8x^{3} -20x^{2}(fg)(x)=2x−5x2=2x−5x2\left(\frac{f}{g}\right)(x) = \frac{2x -5}{x^{2}} = \frac{}{2x} - \frac{5}{x^{2}}For the third part, we have(f+g)(x)=x−+x=x+x−(f+g)(x) = x - + \sqrt{x} = \sqrt{x} + x -(f−g)(x)=x−−x=−x+x−(f-g)(x) = x - - \sqrt{x} = -\sqrt{x} + x -(fg)(x)=(x−)⋅x=xx−x(fg)(x) = (x -) \cdot \sqrt{x} = x\sqrt{x} -\sqrt{x}(fg)(x)=x−x=x−x\left(\frac{f}{g}\right)(x) = \frac{x -}{\sqrt{x}} = \sqrt{x} - \frac{}{\sqrt{x}}

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