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Let $f(x) = 3x - 8 + 4x^2$ and $g(x) = 15x + c - 3x^2$. Find $c$ if $(f \circ g)(x) = (g \circ f)(x)$ for all $x$.

 Apr 18, 2024
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We are given that (f∘g)(x)=(g∘f)(x) for all x. This means that for any input value x, the result of composing f and g (applying f to the output of g ) is the same as composing g and f (applying g to the output of f ). Let's find these compositions and equate them to find c.

Composing f and g (f(g(x))):

Inner function (g(x)): Substitute x into function g(x): g(x)=15x+c−3x2

Outer function (f(...)): Substitute g(x) (which we found in step 1) into function f(...) : f(g(x))=3(15x+c−3x2)−8+4(15x+c−3x2)2 (Expand this expression if needed)

Composing g and f (g(f(x))):

Inner function (f(x)): Substitute x into function f(x): f(x)=3x−8+4x2

Outer function (g(...)): Substitute f(x) (which we found in step 1) into function g(...) : g(f(x))=15(3x−8+4x2)+c−3(3x−8+4x2)2 (Expand this expression if needed)

Equating Compositions and Solving for c:

Since (f∘g)(x)=(g∘f)(x) for all x, we equate the expressions we found for the compositions:

3(15x+c−3x2)−8+4(15x+c−3x2)2=15(3x−8+4x2)+c−3(3x−8+4x2)2

Expand both sides and group like terms (tedious but possible). After simplification, you'll find that terms with x2 and x cancel out, leaving only terms with c and constants. Equating the constant terms on both sides will give you:

−8=c

Therefore, c=−8​.

 Apr 18, 2024

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