Math  /  Algebra

QuestionFind f+g,fg,fgf+g, f-g, f g, and f/gf / g and their domains. f(x)=xg(x)=4xf(x)=x \quad g(x)=4 x
Find (f+g)(x)(f+g)(x). \square
Find the domain of (f+g)(x)(f+g)(x). (Enter your answer using interval notation.) \square
Find (fg)(x)(f-g)(x). \square
Find the domain of (fg)(x)(f-g)(x). (Enter your answer using interval notation.) \square
Find (fg)(x)(f g)(x). \square
Find the domain of (fg)(x)(\operatorname{fg})(x). (Enter your answer using interval notation.) \square
Find (fg)(x)\left(\frac{f}{g}\right)(x). \square
Find the domain of (fg)(x)\left(\frac{f}{g}\right)(x). (Enter your answer using interval notation.) \square

Studdy Solution

STEP 1

What is this asking? We're playing with two simple functions, f(x)=xf(x) = x and g(x)=4xg(x) = 4x, combining them in different ways (adding, subtracting, multiplying, and dividing) and figuring out where these new combined functions make sense! Watch out! Be careful when dividing – we can't divide by zero, so we need to exclude any x-values that make the denominator zero.

STEP 2

1. Find f+gf + g and its domain.
2. Find fgf - g and its domain.
3. Find fgfg and its domain.
4. Find f/gf/g and its domain.

STEP 3

Alright, let's **add** the functions! (f+g)(x)(f+g)(x) means f(x)+g(x)f(x) + g(x).
We know f(x)=xf(x) = x and g(x)=4xg(x) = 4x, so (f+g)(x)=x+4x=5x(f+g)(x) = x + 4x = 5x.
Boom!

STEP 4

The **domain** is all the x-values where the function is happy.
Since 5x5x is defined for all real numbers, the domain of (f+g)(x)(f+g)(x) is (,)(-\infty, \infty).
Easy peasy!

STEP 5

Time to **subtract**! (fg)(x)(f-g)(x) means f(x)g(x)f(x) - g(x), which is x4x=3xx - 4x = -3x.
Done!

STEP 6

Just like before, 3x-3x is defined for all real numbers, so the domain of (fg)(x)(f-g)(x) is also (,)(-\infty, \infty).

STEP 7

Now we **multiply**! (fg)(x)(fg)(x) means f(x)g(x)f(x) \cdot g(x), which is x4x=4x2x \cdot 4x = 4x^2.
Nailed it!

STEP 8

4x24x^2 is also defined for all real numbers, so the domain of (fg)(x)(fg)(x) is (,)(-\infty, \infty).

STEP 9

Let's **divide**! (f/g)(x)(f/g)(x) means f(x)g(x)\frac{f(x)}{g(x)}, which is x4x\frac{x}{4x}.
Since xx isn't zero, we can divide both the numerator and denominator by xx to get 14\frac{1}{4}.

STEP 10

Remember, we can't divide by zero!
So, we need to exclude any x-value that makes g(x)=0g(x) = 0.
Since g(x)=4xg(x) = 4x, we set 4x=04x = 0 and find x=0x = 0.
So, the domain of (f/g)(x)(f/g)(x) is all real numbers *except* 0, which is (,0)(0,)(-\infty, 0) \cup (0, \infty).

STEP 11

(f+g)(x)=5x(f+g)(x) = 5x, Domain: (,)(-\infty, \infty) (fg)(x)=3x(f-g)(x) = -3x, Domain: (,)(-\infty, \infty) (fg)(x)=4x2(fg)(x) = 4x^2, Domain: (,)(-\infty, \infty) (f/g)(x)=14(f/g)(x) = \frac{1}{4}, Domain: (,0)(0,)(-\infty, 0) \cup (0, \infty)

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