Find each of the following for [tex]f(x)=4 x^2-6 x+5[/tex]
(a) [tex]f(x+h)[/tex]
(b) [tex]f(x+h)-f(x)[/tex]
(c) [tex]\frac{f(x+h)-f(x)}{h}[/tex]
Answer (1)
To find f ( x + h ) for the function f ( x ) = 4 x 2 − 6 x + 5 , we perform the following steps:
Substitute x + h for x in the function: f ( x + h ) = 4 ( x + h ) 2 − 6 ( x + h ) + 5 .
Expand the expression: f ( x + h ) = 4 ( x 2 + 2 x h + h 2 ) − 6 ( x + h ) + 5 .
Distribute and simplify: f ( x + h ) = 4 x 2 + 8 x h + 4 h 2 − 6 x − 6 h + 5 .
The final expression is: 4 x 2 + 8 x h + 4 h 2 − 6 x − 6 h + 5 .
Explanation
Find f(x+h) We are given the function f ( x ) = 4 x 2 − 6 x + 5 , and we need to find f ( x + h ) . This means we need to substitute x + h for x in the expression for f ( x ) .
Expanding the expression So, we have f ( x + h ) = 4 ( x + h ) 2 − 6 ( x + h ) + 5 . Now, let's expand and simplify this expression. First, expand ( x + h ) 2 to get x 2 + 2 x h + h 2 .
Substituting and distributing Now, substitute this back into the expression for f ( x + h ) :
f ( x + h ) = 4 ( x 2 + 2 x h + h 2 ) − 6 ( x + h ) + 5
Distribute the 4 and the -6:
f ( x + h ) = 4 x 2 + 8 x h + 4 h 2 − 6 x − 6 h + 5
Final expression for f(x+h) So, f ( x + h ) = 4 x 2 + 8 x h + 4 h 2 − 6 x − 6 h + 5 . This is our final expression for f ( x + h ) .
Examples
Understanding function transformations is crucial in many fields. For example, in physics, if you know the position of an object as a function of time, f ( t ) , then f ( t + h ) would represent the position of the object at a time h later. Similarly, in economics, if f ( x ) represents the cost of producing x items, then f ( x + h ) would represent the cost of producing h more items. This concept is also used in computer graphics for animation and transformations.
