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Math  /  Algebra

QuestionThe functions ff and gg are defined f(x)=x2−x−4 and g(x)=x−39x−5f(x)=x^{2}-x-4 \quad \text { and } \quad g(x)=\frac{x-3}{9 x-5}
Find f(x+8)f(x+8) and g(x4)g\left(\frac{x}{4}\right). Write your answers without pare possible. f(x+8)=g(x4)=□\begin{array}{l} f(x+8)= \\ g\left(\frac{x}{4}\right)=\square \end{array}

Studdy Solution

STEP 1

1. We are given two functions f(x) f(x) and g(x) g(x) .
2. We need to find f(x+8) f(x+8) by substituting x+8 x+8 into f(x) f(x) .
3. We need to find g(x4) g\left(\frac{x}{4}\right) by substituting x4 \frac{x}{4} into g(x) g(x) .

STEP 2

1. Substitute x+8 x+8 into the function f(x) f(x) and simplify.
2. Substitute x4 \frac{x}{4} into the function g(x) g(x) and simplify.

STEP 3

Substitute x+8 x+8 into f(x)=x2−x−4 f(x) = x^2 - x - 4 .
f(x+8)=(x+8)2−(x+8)−4 f(x+8) = (x+8)^2 - (x+8) - 4

STEP 4

Expand (x+8)2 (x+8)^2 .
(x+8)2=x2+16x+64 (x+8)^2 = x^2 + 16x + 64

STEP 5

Substitute the expanded form back into the expression:
f(x+8)=x2+16x+64−x−8−4 f(x+8) = x^2 + 16x + 64 - x - 8 - 4

STEP 6

Combine like terms:
f(x+8)=x2+15x+52 f(x+8) = x^2 + 15x + 52

STEP 7

Substitute x4 \frac{x}{4} into g(x)=x−39x−5 g(x) = \frac{x-3}{9x-5} .
g(x4)=x4−39(x4)−5 g\left(\frac{x}{4}\right) = \frac{\frac{x}{4} - 3}{9\left(\frac{x}{4}\right) - 5}

STEP 8

Simplify the numerator:
x4−3=x−124 \frac{x}{4} - 3 = \frac{x - 12}{4}

STEP 9

Simplify the denominator:
9(x4)−5=9x4−5=9x−204 9\left(\frac{x}{4}\right) - 5 = \frac{9x}{4} - 5 = \frac{9x - 20}{4}

STEP 10

Combine the simplified numerator and denominator:
g(x4)=x−1249x−204=x−129x−20 g\left(\frac{x}{4}\right) = \frac{\frac{x - 12}{4}}{\frac{9x - 20}{4}} = \frac{x - 12}{9x - 20}
The results are: f(x+8)=x2+15x+52 f(x+8) = x^2 + 15x + 52 g(x4)=x−129x−20 g\left(\frac{x}{4}\right) = \frac{x - 12}{9x - 20}

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