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Math  /  Algebra

QuestionLet f(x)=3−xf(x)=\sqrt{3-x} and g(x)=16−x2g(x)=\sqrt{16-x^{2}}. Find f+g,f−g,f⋅gf+g, f-g, f \cdot g, and fg\frac{f}{g}, and their respective domains.
1. f+g=f+g= □\square
2. What is the domain of f+gf+g ?
Answer (in interval notation): □\square
3. f−g=f-g= □\square
4. What is the domain of f−gf-g ?
Answer (in interval notation): □\square
5. f⋅g=f \cdot g= □\square
6. What is the domain of f⋅gf \cdot g ?
Answer (in interval notation): □\square
7. fg=\frac{f}{g}= □\square
8. What is the domain of fg\frac{f}{g} ?
Answer (in interval notation): □\square

Studdy Solution

STEP 1

1. The function f(x)=3−x f(x) = \sqrt{3-x} is defined where the radicand 3−x≥0 3-x \geq 0 .
2. The function g(x)=16−x2 g(x) = \sqrt{16-x^2} is defined where the radicand 16−x2≥0 16-x^2 \geq 0 .
3. The domain of any combination of functions is the intersection of the domains of the individual functions, considering any additional restrictions.

STEP 2

1. Determine the domain of f(x) f(x) .
2. Determine the domain of g(x) g(x) .
3. Find f+g f+g and its domain.
4. Find f−g f-g and its domain.
5. Find f⋅g f \cdot g and its domain.
6. Find fg \frac{f}{g} and its domain.

STEP 3

Determine the domain of f(x)=3−x f(x) = \sqrt{3-x} :
3−x≥0 3-x \geq 0 x≤3 x \leq 3
Thus, the domain of f(x) f(x) is (−∞,3] (-\infty, 3] .

STEP 4

Determine the domain of g(x)=16−x2 g(x) = \sqrt{16-x^2} :
16−x2≥0 16-x^2 \geq 0 x2≤16 x^2 \leq 16
This implies:
−4≤x≤4 -4 \leq x \leq 4
Thus, the domain of g(x) g(x) is [−4,4] [-4, 4] .

STEP 5

Find f+g f+g :
f+g=3−x+16−x2 f+g = \sqrt{3-x} + \sqrt{16-x^2}
The domain of f+g f+g is the intersection of the domains of f f and g g :
(−∞,3]∩[−4,4]=[−4,3] (-\infty, 3] \cap [-4, 4] = [-4, 3]

STEP 6

Find f−g f-g :
f−g=3−x−16−x2 f-g = \sqrt{3-x} - \sqrt{16-x^2}
The domain of f−g f-g is the same as the domain of f+g f+g , which is:
[−4,3] [-4, 3]

STEP 7

Find f⋅g f \cdot g :
f⋅g=3−x⋅16−x2 f \cdot g = \sqrt{3-x} \cdot \sqrt{16-x^2}
The domain of f⋅g f \cdot g is the same as the domain of f+g f+g , which is:
[−4,3] [-4, 3]

STEP 8

Find fg \frac{f}{g} :
fg=3−x16−x2 \frac{f}{g} = \frac{\sqrt{3-x}}{\sqrt{16-x^2}}
The domain of fg \frac{f}{g} must exclude points where g(x)=0 g(x) = 0 . Since g(x)=16−x2 g(x) = \sqrt{16-x^2} , it is zero when x=±4 x = \pm 4 .
Thus, the domain of fg \frac{f}{g} is:
(−4,3] (-4, 3]

1. f+g=3−x+16−x2 f+g = \sqrt{3-x} + \sqrt{16-x^2}
2. Domain of f+g f+g : [−4,3][-4, 3]
3. f−g=3−x−16−x2 f-g = \sqrt{3-x} - \sqrt{16-x^2}
4. Domain of f−g f-g : [−4,3][-4, 3]
5. f⋅g=3−x⋅16−x2 f \cdot g = \sqrt{3-x} \cdot \sqrt{16-x^2}
6. Domain of f⋅g f \cdot g : [−4,3][-4, 3]
7. fg=3−x16−x2 \frac{f}{g} = \frac{\sqrt{3-x}}{\sqrt{16-x^2}}
8. Domain of fg \frac{f}{g} : (−4,3](-4, 3]

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