Let [tex]$f(x)=4 x^2-2$[/tex]. Find [tex]$f(t+1)$[/tex].
Answer (1)
Substitute ( t + 1 ) for x in the function f ( x ) = 4 x 2 − 2 .
Expand the expression: f ( t + 1 ) = 4 ( t + 1 ) 2 − 2 = 4 ( t 2 + 2 t + 1 ) − 2 .
Distribute and simplify: f ( t + 1 ) = 4 t 2 + 8 t + 4 − 2 .
The final expression is: 4 t 2 + 8 t + 2 .
Explanation
Understanding the problem We are given the function f ( x ) = 4 x 2 − 2 and asked to find f ( t + 1 ) . This means we need to substitute ( t + 1 ) for x in the function's expression.
Substitution Now, let's substitute ( t + 1 ) for x in the expression for f ( x ) : f ( t + 1 ) = 4 ( t + 1 ) 2 − 2
Expanding the square Next, we expand the term ( t + 1 ) 2 :
( t + 1 ) 2 = ( t + 1 ) ( t + 1 ) = t 2 + t + t + 1 = t 2 + 2 t + 1
Substituting back Now, substitute this back into the expression for f ( t + 1 ) :
f ( t + 1 ) = 4 ( t 2 + 2 t + 1 ) − 2
Distributing Distribute the 4: f ( t + 1 ) = 4 t 2 + 8 t + 4 − 2
Simplifying Finally, combine the constant terms: f ( t + 1 ) = 4 t 2 + 8 t + 2
Examples
Understanding function transformations is crucial in many fields. For instance, in physics, if you have a formula that describes the position of an object at time x , f ( x ) , then f ( x + 1 ) would describe the position of the object one unit of time later. Similarly, in economics, if f ( x ) represents the cost of producing x items, then f ( x + 1 ) would represent the cost of producing one additional item. This type of function evaluation and simplification is a fundamental skill.
