Studdy AI Logo
Math  /  Algebra

QuestionFind (f∘g)(x)(f \circ g)(x) and (g∘f)(x)(g \circ f)(x) and the domain of each. f(x)=x+3,g(x)=2x2−5x−3f(x)=x+3, g(x)=2 x^{2}-5 x-3 (f∘g)(x)=(f \circ g)(x)= □\square (Simplify your answer.)
The domain of (f∘g)(x)(f \circ g)(x) is □\square . (Type your answer in interval notation.) (g∘f)(x)=(g \circ f)(x)= □\square (Simplify your answer.)
The domain of (g∘f)(x)(g \circ f)(x) is □\square . (Type your answer in interval notation.)

Studdy Solution

STEP 1

1. We are given two functions f(x)=x+3 f(x) = x + 3 and g(x)=2x2−5x−3 g(x) = 2x^2 - 5x - 3 .
2. We need to find the compositions (f∘g)(x) (f \circ g)(x) and (g∘f)(x) (g \circ f)(x) .
3. The domain of a composite function (f∘g)(x) (f \circ g)(x) is determined by the domain of g(x) g(x) and the domain of f(g(x)) f(g(x)) .
4. Similarly, the domain of (g∘f)(x) (g \circ f)(x) is determined by the domain of f(x) f(x) and the domain of g(f(x)) g(f(x)) .

STEP 2

1. Find (f∘g)(x) (f \circ g)(x) .
2. Determine the domain of (f∘g)(x) (f \circ g)(x) .
3. Find (g∘f)(x) (g \circ f)(x) .
4. Determine the domain of (g∘f)(x) (g \circ f)(x) .

STEP 3

To find (f∘g)(x) (f \circ g)(x) , substitute g(x) g(x) into f(x) f(x) :
(f∘g)(x)=f(g(x))=f(2x2−5x−3) (f \circ g)(x) = f(g(x)) = f(2x^2 - 5x - 3)
Now substitute into f(x)=x+3 f(x) = x + 3 :
f(2x2−5x−3)=(2x2−5x−3)+3 f(2x^2 - 5x - 3) = (2x^2 - 5x - 3) + 3
Simplify the expression:
=2x2−5x−3+3 = 2x^2 - 5x - 3 + 3 =2x2−5x = 2x^2 - 5x

STEP 4

The domain of (f∘g)(x) (f \circ g)(x) is determined by the domain of g(x) g(x) since f(x)=x+3 f(x) = x + 3 is defined for all real numbers. The function g(x)=2x2−5x−3 g(x) = 2x^2 - 5x - 3 is a polynomial, which is defined for all real numbers.
Thus, the domain of (f∘g)(x) (f \circ g)(x) is:
(−∞,∞) (-\infty, \infty)

STEP 5

To find (g∘f)(x) (g \circ f)(x) , substitute f(x) f(x) into g(x) g(x) :
(g∘f)(x)=g(f(x))=g(x+3) (g \circ f)(x) = g(f(x)) = g(x + 3)
Now substitute into g(x)=2x2−5x−3 g(x) = 2x^2 - 5x - 3 :
g(x+3)=2(x+3)2−5(x+3)−3 g(x + 3) = 2(x + 3)^2 - 5(x + 3) - 3
Expand and simplify:
=2(x2+6x+9)−5x−15−3 = 2(x^2 + 6x + 9) - 5x - 15 - 3 =2x2+12x+18−5x−15−3 = 2x^2 + 12x + 18 - 5x - 15 - 3 =2x2+7x = 2x^2 + 7x

STEP 6

The domain of (g∘f)(x) (g \circ f)(x) is determined by the domain of f(x) f(x) since g(x)=2x2−5x−3 g(x) = 2x^2 - 5x - 3 is defined for all real numbers. The function f(x)=x+3 f(x) = x + 3 is defined for all real numbers.
Thus, the domain of (g∘f)(x) (g \circ f)(x) is:
(−∞,∞) (-\infty, \infty)
The simplified expressions and domains are: - (f∘g)(x)=2x2−5x (f \circ g)(x) = 2x^2 - 5x - The domain of (f∘g)(x) (f \circ g)(x) is (−∞,∞) (-\infty, \infty) . - (g∘f)(x)=2x2+7x (g \circ f)(x) = 2x^2 + 7x - The domain of (g∘f)(x) (g \circ f)(x) is (−∞,∞) (-\infty, \infty) .

Was this helpful?

Studdy solves anything!

banner

Start learning now

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

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord