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Asked: 2023-10-28T03:28:06+00:00 In:

If f(x)=x^{2} + 1, g(x)=2x-3 and h(x)=4x+5 Find (i) f[g(h(x))] (ii) h[g(f(x))]

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If f(x)=x^{2} + 1, g(x)=2x-3) and h(x)=4x+5

Find

i) f[g(h(x))]

ii) h[g(f(x))]

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1 Answer

  1. charles
    Begginer
    This answer was edited.

    Given

    f(x)=x^{2}+1, \: g(x)=2x-3 and h(x)=4x+5

    i) For f[g(h(x))]

    To solve this, we have to substitute the value of h(x) into g(x) and then substitute the result into f(x).

    Let’s substitute h(x) into g(x)

    g(h(x))=g(4x+5)

    Next, substitute the above into f(x)

    f[g(h(x))] = f[g(4x + 5)] = f(2(4x + 5) - 3)

    = f(8x+10-3)

    = f(8x+7)

    Recall that f(x)=x^{2}+1

    f(8x+7)=(8x+7)^{2}+1

    Expanding;

    =64x^{2}+112x+50

    \therefore f[g(h(x))]=64x^{2}+112x+50

    ii) For h[g(f(x))]

    Just like we did in the first question, we begin by substituting f(x) into g(x) and then substitute the result into h(x).

    Let’s substitute f(x) into g(x)

    g(f(x))=g(x^{2}+1)

    Next, substitute the above into h(x)

    h[g(f(x))]=h[g(x^{2}+1)]=h(2(x^{2}+1)-3)

    =h(2x^{2}+2-3)

    =h(2x^{2}-1)

    Recall that h(x)=4x+5

    \therefore h(2x^{2}-1) = 4(2x^{2}-1)+5

    =8x^{2}-4+5

    =8x^{2}+1

    \therefore h[g(f(x))]=8x^{2}+1

     

     

     

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