$f(x)=x+3$ and $g(x)=x-3$
(a) $f(g(x))=$ $\square$
(b) $g(f(x))=$ $\square$
(c) Thus $g(x)$ is called an $\square$ function of $f(x)
Answer (2)
Calculate f ( g ( x )) by substituting g ( x ) into f ( x ) , resulting in f ( g ( x )) = x .
Calculate g ( f ( x )) by substituting f ( x ) into g ( x ) , resulting in g ( f ( x )) = x .
Since f ( g ( x )) = g ( f ( x )) = x , conclude that g ( x ) is the inverse function of f ( x ) .
The final answer is: f ( g ( x )) = x , g ( f ( x )) = x , and g ( x ) is the inverse function of f ( x ) . f ( g ( x )) = x , g ( f ( x )) = x , inverse
Explanation
Understanding the Problem We are given two functions, f ( x ) = x + 3 and g ( x ) = x − 3 . We need to find the composite functions f ( g ( x )) and g ( f ( x )) , and then determine the relationship between f ( x ) and g ( x ) .
Finding f(g(x)) To find f ( g ( x )) , we substitute g ( x ) into f ( x ) . So, f ( g ( x )) = f ( x − 3 ) = ( x − 3 ) + 3 .
Simplifying f(g(x)) Simplifying the expression, we get f ( g ( x )) = x − 3 + 3 = x .
Finding g(f(x)) To find g ( f ( x )) , we substitute f ( x ) into g ( x ) . So, g ( f ( x )) = g ( x + 3 ) = ( x + 3 ) − 3 .
Simplifying g(f(x)) Simplifying the expression, we get g ( f ( x )) = x + 3 − 3 = x .
Determining the Relationship Since f ( g ( x )) = x and g ( f ( x )) = x , we can conclude that g ( x ) is the inverse function of f ( x ) .
Examples
Understanding function composition is crucial in many real-world applications. For instance, consider a store that applies a discount and then adds sales tax. If f ( x ) represents the discount function and g ( x ) represents the sales tax function, then f ( g ( x )) calculates the final price after tax is applied to the discounted price. Similarly, g ( f ( x )) calculates the final price if the discount is applied after the tax. In this case, since f ( g ( x )) = g ( f ( x )) = x , it means that applying the discount and then the tax, or vice versa, results in the same final price, simplifying the pricing strategy.
To find f ( g ( x )) , we substitute g ( x ) into f ( x ) and get f ( g ( x )) = x . Similarly, by substituting f ( x ) into g ( x ) , we get g ( f ( x )) = x . This shows that g ( x ) is the inverse function of f ( x ) .
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