Determine whether each function graphed or defined is one-to-one. y = 5|x+2|
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- 0. Review of Algebra4h 18m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations1h 43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 5m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 22m
- 10. Combinatorics & Probability1h 45m
3. Functions
Function Composition
Problem 43
Textbook Question
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 guidance1
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.
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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.
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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.
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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.
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