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QuestionEvaluate the functions: 1. (h∘g)(x)(h \circ g)(x) and 2. 2⋅[f∘(g∘g)](x)2 \cdot [f \circ (g \circ g)](x), where g(x)=3x−2g(x)=3x-2, f(x)=−xf(x)=\sqrt{-x}, h(x)=3x+4h(x)=\frac{3}{x+4}.

Studdy Solution

STEP 1

Assumptions1. The functions are defined as follows - g(x)=3x−g(x)=3x- - f(x)=−xf(x)=\sqrt{-x} - h(x)=3x+4h(x)=\frac{3}{x+4} . The composite functions are defined as follows - h∘g(x)h \circ g(x) - f∘(g∘g)(x)f \circ(g \circ g)(x)3. The operations to be performed are - Evaluate 1⋅(h∘g)(x)1 \cdot(h \circ g)(x) - Evaluate ⋅[f∘(g∘g)(x)] \cdot[f \circ(g \circ g)(x)]

STEP 2

First, let's evaluate the composite function h∘g(x)h \circ g(x). This means we substitute g(x)g(x) into h(x)h(x).
h∘g(x)=h(g(x))=g(x)+4h \circ g(x) = h(g(x)) = \frac{}{g(x)+4}

STEP 3

Now, substitute g(x)=3x−2g(x) =3x -2 into the above equation.
h∘g(x)=33x−2+h \circ g(x) = \frac{3}{3x -2 +}

STEP 4

implify the equation.
h∘g(x)=33x+2h \circ g(x) = \frac{3}{3x +2}

STEP 5

Now, evaluate 1⋅(h∘g)(x)1 \cdot(h \circ g)(x) by multiplying the result of h∘g(x)h \circ g(x) by1.
1⋅(h∘g)(x)=1⋅33x+2=33x+21 \cdot(h \circ g)(x) =1 \cdot \frac{3}{3x +2} = \frac{3}{3x +2}

STEP 6

Next, let's evaluate the composite function f∘(g∘g)(x)f \circ(g \circ g)(x). This means we substitute g(x)g(x) into g(x)g(x) first, and then substitute the result into f(x)f(x).
f∘(g∘g)(x)=f(g(g(x)))f \circ(g \circ g)(x) = f(g(g(x)))

STEP 7

Substitute g(x)=3x−2g(x) =3x -2 into g(x)g(x).
g(g(x))=g(3x−2)=3(3x−2)−2=9x−6−2=9x−g(g(x)) = g(3x -2) =3(3x -2) -2 =9x -6 -2 =9x -

STEP 8

Now, substitute g(g(x))=x−8g(g(x)) =x -8 into f(x)f(x).
f∘(g∘g)(x)=f(x−8)=−(x−8)f \circ(g \circ g)(x) = f(x -8) = \sqrt{-(x -8)}

STEP 9

implify the equation.
f∘(g∘g)(x)=−9x+8f \circ(g \circ g)(x) = \sqrt{-9x +8}

STEP 10

Finally, evaluate 2⋅[f∘(g∘g)(x)]2 \cdot[f \circ(g \circ g)(x)] by multiplying the result of f∘(g∘g)(x)f \circ(g \circ g)(x) by2.
2⋅[f∘(g∘g)(x)]=2⋅−9x+82 \cdot[f \circ(g \circ g)(x)] =2 \cdot \sqrt{-9x +8}So, the solutions are. ⋅(h∘g)(x)=33x+2 \cdot(h \circ g)(x) = \frac{3}{3x +2}
2. 2⋅[f∘(g∘g)(x)]=2⋅−9x+82 \cdot[f \circ(g \circ g)(x)] =2 \cdot \sqrt{-9x +8}

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