Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If $f of x equals 3 x$ and $g (x) = x + 5$ , find ${(f \circ g)}^{- 1} (x)$ and $(g^{- 1} \circ f^{- 1}) (x)$ .

검증된 단계별 안내

First, understand the notation: (ƒ 0 g)(x) means the composition of functions f and g, which is f(g(x)). Similarly, (ƒ 0 g)^{-1}(x) means the inverse of the composition f(g(x)).

Step 1: Find the composition (ƒ 0 g)(x) by substituting g(x) into f. Since f(x) = 3x and g(x) = x + 5, write the expression for f(g(x)) as $f(g(x)) = 3(x + 5)$.

Step 2: To find the inverse of the composition, set $y = 3(x + 5)$ and solve for x in terms of y. This involves isolating x on one side of the equation.

Step 3: Next, find the inverse functions individually: find $f^{-1}(x)$ by solving $y = 3x$ for x, and find $g^{-1}(x)$ by solving $y = x + 5$ for x.

Step 4: Finally, find the composition $(g^{-1} 0 f^{-1})(x)$ by substituting $f^{-1}(x)$ into $g^{-1}$. Write the expression for $g^{-1}(f^{-1}(x))$.

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주요 개념

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Function Composition

Function composition involves applying one function to the result of another, denoted as (f ∘ g)(x) = f(g(x)). Understanding how to combine functions correctly is essential for evaluating expressions like (f ∘ g)(x) and manipulating them for further operations.

Inverse Functions

An inverse function reverses the effect of the original function, such that f(f⁻¹(x)) = x. Finding the inverse requires solving for x in terms of y and swapping variables. Recognizing and computing inverses is crucial for expressions involving f⁻¹ or g⁻¹.

Notation and Order of Operations in Compositions and Inverses

Understanding the notation (f ∘ g)⁻¹ and (g⁻¹ ∘ f⁻¹) requires knowing that the inverse of a composition reverses the order: (f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹. This concept helps correctly interpret and simplify composite inverse functions.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.If f(x)=3xf(x) = 3x and g(x)=x+5g(x) = x + 5, find (f∘g)−1(x)\(\left\)(f\(\circ\) g\(\right\))^{-1}(x) and (g−1∘f−1)(x)(g^{-1} \(\circ\) f^{-1})(x).