for $f(x) = x^2$ and $g(x) = x^2 + 1$, find the following composite functions and state the domain of each.
(a) $f \circ g$
(b) $g \circ f$
(c) $f \circ f$
(d) $g \circ g$

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Composite Functions", "Domain and Range", "Interval Notation" ], "new_concepts": [], "current_concepts": [ "Composite Functions", "Domain and Range", "Interval Notation" ] } ## Find the composite function (f ∘ g)(x) and its domain Using the *Composite Functions* and *Domain and Range* knowledge points $$ \begin{aligned} (f \circ g)(x) &= f(g(x)) = f(x^2 + 1) = (x^2 + 1)^2 = x^4 + 2x^2 + 1\ \text{Domain: } &(-\infty, \infty) \end{aligned} $$ ## Find the composite function (g ∘ f)(x) and its domain Using the *Composite Functions* and *Domain and Range* knowledge points $$ \begin{aligned} (g \circ f)(x) &= g(f(x)) = g(x^2) = (x^2)^2 + 1 = x^4 + 1\ \text{Domain: } &(-\infty, \infty) \end{aligned} $$ ## Find the composite function (f ∘ f)(x) and its domain Using the *Composite Functions* and *Domain and Range* knowledge points $$ \begin{aligned} (f \circ f)(x) &= f(f(x)) = f(x^2) = (x^2)^2 = x^4\ \text{Domain: } &(-\infty, \infty) \end{aligned} $$ ## Find the composite function (g ∘ g)(x) and its domain Using the *Composite Functions* and *Domain and Range* knowledge points $$ \begin{aligned} (g \circ g)(x) &= g(g(x)) = g(x^2 + 1) = (x^2 + 1)^2 + 1 = x^4 + 2x^2 + 2\ \text{Domain: } &(-\infty, \infty) \end{aligned} $$ (a) $(f \circ g)(x) = x^4 + 2x^2 + 1$; Domain: $(-\infty, \infty)$ (b) $(g \circ f)(x) = x^4 + 1$; Domain: $(-\infty, \infty)$ (c) $(f \circ f)(x) = x^4$; Domain: $(-\infty, \infty)$ (d) $(g \circ g)(x) = x^4 + 2x^2 + 2$; Domain: $(-\infty, \infty)$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Composite Functions" ] }

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Find the composite function (f ∘ g)(x) and its domain

Find the composite function (g ∘ f)(x) and its domain

Find the composite function (f ∘ f)(x) and its domain

Find the composite function (g ∘ g)(x) and its domain

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