If [tex]f(x)=x+2[/tex] and [tex]g(x)=x^2-3[/tex], find:
i) [tex]f(g(x))[/tex]
ii) [tex]g(f(x))[/tex]
Answer (1)
To find f ( g ( x )) , substitute g ( x ) into f ( x ) and simplify: f ( g ( x )) = f ( x 2 − 3 ) = ( x 2 − 3 ) + 2 = x 2 − 1 .
To find g ( f ( x )) , substitute f ( x ) into g ( x ) and simplify: g ( f ( x )) = g ( x + 2 ) = ( x + 2 ) 2 − 3 = x 2 + 4 x + 4 − 3 = x 2 + 4 x + 1 .
Thus, f ( g ( x )) = x 2 − 1 .
And, g ( f ( x )) = x 2 + 4 x + 1 .
f ( g ( x )) = x 2 − 1 , g ( f ( x )) = x 2 + 4 x + 1
Explanation
Understanding the Problem We are given two functions, f ( x ) = x + 2 and g ( x ) = x 2 − 3 . We need to find the composite functions f ( g ( x )) and g ( f ( x )) .
Finding f(g(x)) To find f ( g ( x )) , we substitute g ( x ) into f ( x ) . This means we replace every instance of x in f ( x ) with g ( x ) . So, f ( g ( x )) = f ( x 2 − 3 ) = ( x 2 − 3 ) + 2 .
Simplifying f(g(x)) Now, we simplify the expression for f ( g ( x )) : f ( g ( x )) = x 2 − 3 + 2 = x 2 − 1
Finding g(f(x)) To find g ( f ( x )) , we substitute f ( x ) into g ( x ) . This means we replace every instance of x in g ( x ) with f ( x ) . So, g ( f ( x )) = g ( x + 2 ) = ( x + 2 ) 2 − 3 .
Simplifying g(f(x)) Now, we simplify the expression for g ( f ( x )) : g ( f ( x )) = ( x + 2 ) 2 − 3 = ( x 2 + 4 x + 4 ) − 3 = x 2 + 4 x + 1
Final Answer Therefore, f ( g ( x )) = x 2 − 1 and g ( f ( x )) = x 2 + 4 x + 1 .
Examples
Composite functions are used in many real-world applications. For example, in computer graphics, transformations like scaling, rotation, and translation are often represented as functions. Applying a series of these transformations to an object involves composing these functions. Similarly, in calculus, the chain rule is used to find the derivative of a composite function, which is essential in optimization problems and related rates.
