Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = 2/x+6, g(x) = 6x+2/x

Verified step by step guidance

1. Step 1: Understand the definition of inverse functions. Two functions $f$ and $g$ are inverses if $f(g(x))=x$ and $g(f(x))=x$ for all $x$ in the domains of $g$ and $f$ respectively.
2. Step 2: Calculate $f(g(x))$. Substitute $g(x)=\frac{6x+2}{x}$ into $f(x)=\frac{2}{x+6}$. This gives $f(g(x))=\frac{2}{\frac{6x+2}{x}+6}$. Simplify this expression.
3. Step 3: Simplify $f(g(x))$. Combine the terms in the denominator: $\frac{6x+2}{x}+6=\frac{6x+2+6x}{x}=\frac{12x+2}{x}$. Then, $f(g(x))=\frac{2x}{12x+2}$. Simplify further if possible.
4. Step 4: Calculate $g(f(x))$. Substitute $f(x)=\frac{2}{x+6}$ into $g(x)=\frac{6x+2}{x}$. This gives $g(f(x))=\frac{6\left(\frac{2}{x+6}\right)+2}{\frac{2}{x+6}}$. Simplify this expression.
5. Step 5: Simplify $g(f(x))$. Simplify the expression $\frac{6\left(\frac{2}{x+6}\right)+2}{\frac{2}{x+6}}=\frac{\frac{12}{x+6}+2}{\frac{2}{x+6}}$. Simplify further to check if it equals $x$.

Key Concepts

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. If f(x) is a function, its inverse, denoted as f⁻¹(x), satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f⁻¹. To determine if two functions are inverses, we check if f(g(x)) = x and g(f(x)) = x for all x in their respective domains.

Composition of Functions

The composition of functions involves applying one function to the result of another. For functions f and g, the composition is denoted as (f ∘ g)(x) = f(g(x)). This operation is crucial for verifying if two functions are inverses, as it allows us to check if the output returns to the original input.

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values). When determining if two functions are inverses, it is important to consider their domains and ranges, as they must align appropriately for the functions to truly be inverses of each other.