If [tex]h(x)=5+x[/tex] and [tex]k(x)=\frac{1}{x}[/tex], which expression is equivalent to [tex](k \circ h)(x)[/tex]?
A. [tex]\frac{(5+x)}{x}[/tex]
B. [tex]\frac{1}{(5+x)}[/tex]
C. [tex]5+\left(\frac{1}{x}\right)[/tex]
D. [tex]5+(5+x)[/tex]
Answer (2)
We are given h ( x ) = 5 + x and k ( x ) = f r a c 1 x .
We need to find ( k c i rc h ) ( x ) = k ( h ( x )) .
Substitute h ( x ) into k ( x ) : k ( 5 + x ) = f r a c 1 5 + x .
Therefore, ( k c i rc h ) ( x ) = b o x e d f r a c 1 5 + x .
Explanation
Understanding the Problem We are given two functions, h ( x ) = 5 + x and k ( x ) = f r a c 1 x . We need to find the expression that is equivalent to the composition ( k c i rc h ) ( x ) , which means k ( h ( x )) . In other words, we need to substitute h ( x ) into k ( x ) .
Finding the Composition First, we identify h ( x ) , which is given as 5 + x . Next, we need to find k ( h ( x )) , which means we need to evaluate k ( 5 + x ) . Since k ( x ) = f r a c 1 x , we replace x in the expression for k ( x ) with ( 5 + x ) . This gives us k ( 5 + x ) = f r a c 1 5 + x .
Final Result Therefore, ( k c i rc h ) ( x ) = k ( h ( x )) = k ( 5 + x ) = f r a c 1 5 + x .
Examples
Understanding function composition is crucial in many areas, such as physics and computer science. For instance, in physics, you might describe the position of a particle as a function of time, h ( t ) , and then describe the energy of the particle as a function of its position, k ( x ) . The composition ( k c i rc h ) ( t ) would then give you the energy of the particle as a function of time. In computer graphics, transformations like scaling and rotations can be represented as functions, and composing these functions allows you to apply multiple transformations in a specific order. This concept is also used in signal processing, where functions can represent signals, and composition can represent the application of filters or other operations on those signals.
The composition ( k ∘ h ) ( x ) is calculated by substituting h ( x ) = 5 + x into k ( x ) = x 1 . This gives us the result k ( h ( x )) = 5 + x 1 , which corresponds to option B. Thus, the answer is B: 5 + x 1 .
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