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QuestionFind the compositions of functions f(x)=5x+4f(x)=5x+4 and g(x)=x−45g(x)=\frac{x-4}{5}: a. (f∘g)(x)(f \circ g)(x), b. (g∘f)(x)(g \circ f)(x), c. (f∘g)(5)(f \circ g)(5), d. (g∘f)(5)(g \circ f)(5).

Studdy Solution

STEP 1

Assumptions1. The function f(x)=5x+4f(x)=5x+4 . The function g(x)=x−45g(x)=\frac{x-4}{5}
3. The composition of functions is defined as (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)) and (g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x))

STEP 2

First, we need to find (f∘g)(x)(f \circ g)(x). This means we need to substitute g(x)g(x) into the function f(x)f(x).
(f∘g)(x)=f(g(x)) (f \circ g)(x) = f(g(x))

STEP 3

Substitute g(x)g(x) into f(x)f(x).
(f∘g)(x)=f(x−5)=5(x−5)+ (f \circ g)(x) = f\left(\frac{x-}{5}\right) =5\left(\frac{x-}{5}\right) +

STEP 4

implify the expression.
(f∘g)(x)=x−4+4 (f \circ g)(x) = x -4 +4

STEP 5

implify further to get the final expression for (f∘g)(x)(f \circ g)(x).
(f∘g)(x)=x (f \circ g)(x) = x

STEP 6

Next, we need to find (g∘f)(x)(g \circ f)(x). This means we need to substitute f(x)f(x) into the function g(x)g(x).
(g∘f)(x)=g(f(x)) (g \circ f)(x) = g(f(x))

STEP 7

Substitute f(x)f(x) into g(x)g(x).
(g∘f)(x)=g(5x+4)=5x+4−45 (g \circ f)(x) = g(5x+4) = \frac{5x+4-4}{5}

STEP 8

implify the expression.
(g∘f)(x)=5x5 (g \circ f)(x) = \frac{5x}{5}

STEP 9

implify further to get the final expression for (g∘f)(x)(g \circ f)(x).
(g∘f)(x)=x (g \circ f)(x) = x

STEP 10

Now we need to find (f∘g)(5)(f \circ g)(5). This means we need to substitute 55 into the function (f∘g)(x)(f \circ g)(x).
(f∘g)(5)=5 (f \circ g)(5) =5

STEP 11

Finally, we need to find (g∘f)(5)(g \circ f)(5). This means we need to substitute 55 into the function (g∘f)(x)(g \circ f)(x).
(g∘f)(5)=5 (g \circ f)(5) =5 The solutions area. (f∘g)(x)=x(f \circ g)(x) = x b. (g∘f)(x)=x(g \circ f)(x) = x c. (f∘g)(5)=5(f \circ g)(5) =5 d. (g∘f)(5)=5(g \circ f)(5) =5

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