Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=5x-9 and g(x) = (x+5)/9

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First, find the composition \( f(g(x)) \) by substituting \( g(x) = \frac{x+5}{9} \) into \( f(x) = 5x - 9 \). This means replacing every \( x \) in \( f(x) \) with \( \frac{x+5}{9} \). So, write \( f\left(g(x)\right) = 5 \left( \frac{x+5}{9} \right) - 9 \).

Next, simplify the expression for \( f(g(x)) \) by distributing the 5 and combining like terms carefully. This will give you a simplified function in terms of \( x \).

Then, find the composition \( g(f(x)) \) by substituting \( f(x) = 5x - 9 \) into \( g(x) = \frac{x+5}{9} \). Replace every \( x \) in \( g(x) \) with \( 5x - 9 \), so write \( g\left(f(x)\right) = \frac{(5x - 9) + 5}{9} \).

Simplify the expression for \( g(f(x)) \) by combining like terms in the numerator and then dividing by 9 to get a simplified function in terms of \( x \).

Finally, determine whether \( f \) and \( g \) are inverses by checking if both compositions \( f(g(x)) \) and \( g(f(x)) \) simplify to \( x \). If both equal \( x \), then \( f \) and \( g \) are inverse functions of each other.

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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)) or g(f(x)). It requires substituting the entire expression of one function into the variable of the other, allowing us to analyze combined transformations or operations.

Inverse Functions

Two functions f and g are inverses if composing them in either order returns the original input, meaning f(g(x)) = x and g(f(x)) = x. This relationship shows that each function reverses the effect of the other, effectively 'undoing' the transformation.

Algebraic Manipulation

Algebraic manipulation involves simplifying expressions, solving equations, and substituting variables accurately. Mastery of these skills is essential to correctly compute compositions and verify if two functions are inverses by simplifying the resulting expressions.