for $f(x) = 1 - x$ and $g(x) = 4x^2 + x + 3$, find the following functions.
a. $(f \circ g)(x)$, b. $(g \circ f)(x)$, c. $(f \circ g)(3)$, d. $(g \circ f)(3)$

a. $(f \circ g)(x) = -4x^2 - x - 2$
(simplify your answer.)

b. $(g \circ f)(x) = \square$
(simplify your answer.)

Question

for $f(x) = 1 - x$ and $g(x) = 4x^2 + x + 3$, find the following functions.
a. $(f \circ g)(x)$, b. $(g \circ f)(x)$, c. $(f \circ g)(3)$, d. $(g \circ f)(3)$

a. $(f \circ g)(x) = -4x^2 - x - 2$
(simplify your answer.)

b. $(g \circ f)(x) = \square$
(simplify your answer.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Function Composition" ], "new_concepts": [], "current_concepts": [ "Function Composition" ] } ## Evaluate the inner function Using the *Function Composition* knowledge point $$ f(x) = 1 - x $$ ## Substitute the inner function into the outer function Using the *Function Composition* knowledge point $$ \begin{aligned} (g \circ f)(x) &= g(f(x)) \ &= g(1 - x) \ &= 4(1 - x)^2 + (1 - x) + 3 \end{aligned} $$ ## Expand and simplify the expression Using the *Function Composition* knowledge point $$ \begin{aligned} (g \circ f)(x) &= 4(1 - 2x + x^2) + 1 - x + 3 \ &= 4 - 8x + 4x^2 + 4 - x \ &= 4x^2 - 9x + 8 \end{aligned} $$ $ (g \circ f)(x) = $ $4x^2 - 9x + 8$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Function Composition" ] }

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QUESTION IMAGE

Question

Explanation:

Evaluate the inner function

Substitute the inner function into the outer function

Expand and simplify the expression

Answer: