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

Recall that two functions \( f \) and \( g \) are inverses if and only if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the domains of the compositions.

Start by finding the composition \( f(g(x)) \). Substitute \( g(x) = \frac{6x + 2}{x} \) into \( f(x) = \frac{2}{x} + 6 \), so \( f(g(x)) = \frac{2}{\frac{6x + 2}{x}} + 6 \).

Simplify the expression \( f(g(x)) = \frac{2}{\frac{6x + 2}{x}} + 6 \) by multiplying numerator and denominator appropriately to eliminate the complex fraction.

Next, find the composition \( g(f(x)) \). Substitute \( f(x) = \frac{2}{x} + 6 \) into \( g(x) = \frac{6x + 2}{x} \), so \( g(f(x)) = \frac{6\left(\frac{2}{x} + 6\right) + 2}{\frac{2}{x} + 6} \).

Simplify \( g(f(x)) \) by distributing and combining like terms in numerator and denominator, then check if both \( f(g(x)) \) and \( g(f(x)) \) simplify to \( x \). If both do, then \( f \) and \( g \) are inverses.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definition of Inverse Functions

Two functions f and g are inverses if applying one after the other returns the original input, meaning f(g(x)) = x and g(f(x)) = x for all x in their domains. This relationship shows that each function 'undoes' the effect of the other.

Function Composition

Function composition involves substituting one function into another, denoted as (f ∘ g)(x) = f(g(x)). To verify inverses, you compute both f(g(x)) and g(f(x)) and check if both simplify to x, confirming the inverse relationship.

Simplifying Rational Expressions

Simplifying rational expressions requires combining terms, factoring, and reducing fractions to their simplest form. This skill is essential when composing functions like f(x) = 2/(x+6) and g(x) = (6x+2)/x to verify if their compositions equal x.

Recommended video:

Guided course