QuestionJuly 15, 2025

Let f(x)=x^3+9 and g(x)=x^2-8 Find the rule for the function. square You may have misinterpreted the problem. Does gf=g(f(x)) 7

Let f(x)=x^3+9 and g(x)=x^2-8 Find the rule for the function. square You may have misinterpreted the problem. Does gf=g(f(x)) 7
Let f(x)=x^3+9 and g(x)=x^2-8 Find the rule for the function. square You may have misinterpreted the problem. Does gf=g(f(x)) 7

Solution
4.6(308 votes)

Answer

g(f(x)) = x^6 + 18x^3 + 73 Explanation 1. Understand the Composition We need to find g(f(x)), which means substituting f(x) into g(x). 2. Substitute f(x) into g(x) Given f(x) = x^3 + 9, substitute into g(x) = x^2 - 8: g(f(x)) = (x^3 + 9)^2 - 8. 3. Expand the Expression Expand (x^3 + 9)^2: (x^3 + 9)^2 = x^6 + 18x^3 + 81. 4. Simplify the Expression Subtract 8 from the expanded expression: x^6 + 18x^3 + 81 - 8 = x^6 + 18x^3 + 73.

Explanation

1. Understand the Composition<br /> We need to find $g(f(x))$, which means substituting $f(x)$ into $g(x)$.<br /><br />2. Substitute $f(x)$ into $g(x)$<br /> Given $f(x) = x^3 + 9$, substitute into $g(x) = x^2 - 8$: <br /> $g(f(x)) = (x^3 + 9)^2 - 8$.<br /><br />3. Expand the Expression<br /> Expand $(x^3 + 9)^2$: <br /> $(x^3 + 9)^2 = x^6 + 18x^3 + 81$.<br /><br />4. Simplify the Expression<br /> Subtract 8 from the expanded expression: <br /> $x^6 + 18x^3 + 81 - 8 = x^6 + 18x^3 + 73$.
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