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Math  /  Algebra

QuestionFor f(x)=rx+sf(x)=r x+s and g(x)=1r(x−s),r≠0g(x)=\frac{1}{r}(x-s), r \neq 0, find (f∘g)(x)(f \circ g)(x) and (g∘f)(x)(g \circ f)(x). Then determine whether (f∘g)(x)=(g∘f)(x)(f \circ g)(x)=(g \circ f)(x).
What is (f∘g)(x)(f \circ g)(x) ? (f∘g)(x)=(f \circ g)(x)=

Studdy Solution

STEP 1

1. We have two functions: f(x)=rx+s f(x) = r x + s and g(x)=1r(x−s) g(x) = \frac{1}{r}(x - s) .
2. The composition of functions involves substituting one function into another.
3. We need to find the compositions (f∘g)(x) (f \circ g)(x) and (g∘f)(x) (g \circ f)(x) .
4. We will then compare these compositions to check if they are equal.

STEP 2

1. Find the expression for (f∘g)(x) (f \circ g)(x) .
2. Find the expression for (g∘f)(x) (g \circ f)(x) .
3. Compare the expressions to determine if they are equal.

STEP 3

To find (f∘g)(x) (f \circ g)(x) , substitute g(x) g(x) into f(x) f(x) .
Given: f(x)=rx+s f(x) = r x + s g(x)=1r(x−s) g(x) = \frac{1}{r}(x - s)
Substitute g(x) g(x) into f(x) f(x) : (f∘g)(x)=f(g(x))=f(1r(x−s)) (f \circ g)(x) = f\left(g(x)\right) = f\left(\frac{1}{r}(x - s)\right)

STEP 4

Now, compute f(1r(x−s)) f\left(\frac{1}{r}(x - s)\right) :
f(1r(x−s))=r(1r(x−s))+s f\left(\frac{1}{r}(x - s)\right) = r\left(\frac{1}{r}(x - s)\right) + s
Simplify the expression: =(x−s)+s = (x - s) + s
=x = x
Thus, (f∘g)(x)=x (f \circ g)(x) = x .
The expression for (f∘g)(x) (f \circ g)(x) is x \boxed{x} .

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