If f(x) = 3x and g(x) = x + 5, find (ƒ 0 g)¯¹ (x) and (g¯¹ o ƒ˜¹) (x).

Verified step by step guidance

1. <insert step 1> Start by finding the composition of functions $(f\circ g)(x)$, which is $f(g(x))$. Substitute $g(x)=x+5$ into $f(x)=3x$ to get $f(g(x))=3(x+5)$.
2. <insert step 2> Simplify $f(g(x))=3(x+5)$ to get $3x+15$.
3. <insert step 3> To find $(f\circ g{)}^{-1}(x)$, set $y=3x+15$ and solve for $x$.
4. <insert step 4> Rearrange the equation $y=3x+15$ to solve for $x$ by first subtracting 15 from both sides to get $y-15=3x$, then divide by 3 to isolate $x$.
5. <insert step 5> Now, find $g^{-1}(x)$ and $f^{-1}(x)$. For $g(x)=x+5$, solve $y=x+5$ for $x$ to get $g^{-1}(x)=x-5$. For $f(x)=3x$, solve $y=3x$ for $x$ to get $f^{-1}(x)=\frac{y}{3}$.

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. For example, if f(x) and g(x) are two functions, the composition (f o g)(x) means applying g first and then f to the result. Understanding this concept is crucial for solving problems that require evaluating combined functions.

Inverse Functions

An inverse function reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f⁻¹(y) will take y back to x. To find the inverse, you typically solve the equation y = f(x) for x. This concept is essential for determining the inverses of the given functions in the question.

Notation and Operations

Understanding mathematical notation and operations is vital for interpreting and solving expressions involving functions. The notation (ƒ o g)⁻¹(x) indicates the inverse of the composition of f and g, while (g⁻¹ o ƒ⁻¹)(x) represents the composition of the inverses of g and f. Familiarity with these notations helps in accurately performing the required calculations.