g(t) = -2t² - 5t
h(t) = -t + 5
Find (g ∘ h)(x).

To find ( g ∘ h ) ( x ) , we need to compose the functions g ( t ) and h ( t ) . This means we will substitute h ( t ) into g ( t ) wherever there is a t .
Given:

g ( t ) = − 2 t 2
h ( t ) = − t + 5

To find ( g ∘ h ) ( x ) , follow these steps:

Substitute h ( t ) into g ( t ) :
g ( h ( t )) = g ( − t + 5 ) = − 2 ( − t + 5 ) 2

Expand ( − t + 5 ) 2 :
[ (-t + 5)^2 = (-t + 5)(-t + 5)
= (-t)(-t) + 2(-t)(5) + 5^2
= t^2 - 10t + 25 ]

Substitute the expanded form back into the expression for g ( h ( t )) :
[ g(h(t)) = -2(t^2 - 10t + 25)
= -2t^2 + 20t - 50 ]


Thus, the function ( g ∘ h ) ( x ) is − 2 t 2 + 20 t − 50 . This represents the composition of the two functions where we inserted the expression for h ( t ) into the formula for g ( t ) .
Composing functions is a fundamental concept in mathematics, helping us to understand how outputs from one function can feed into another, thereby creating a new function. This technique is widely applied in advanced mathematical computations and problem solving.

Answered by LucasMatthewHarris | 2025-07-06