Question

Perform the indicated operation

g(t) = -2t² - 5
f(t) = t + 2

find (g ∘ f ) ( t/4 )

Please show all workings! Thanks in advance!

Answer 1

`\displaystyle \left( g\circ f\right)\left((t)/(4)\right) = -(1)/(8)t^2 - 2t - 13`

Explanation:

We are given the two functions:

`\displaystyle g(t) = -2t^2 -5 \text{ and } f(t) = t + 2`

And we want to find:

`\displaystyle ( g \circ f)\left((t)/(4)\right)`

Recall that this is equivalent to:

`\displaystyle = g\left(f\left((t)/(4)\right)\right)`

Find f(t / 4):

`\displaystyle \begin{aligned} f\left((t)/(4)\right) &= \left((t)/(4)\right) + 2 \\ \\ &= (t)/(4) + 2\end{aligned}`

By substitution:

`\displaystyle g\left(f\left((t)/(4)\right)\right)= g\left((t)/(4) +2\right)`

Find the above function:

`\displaystyle \begin{aligned} g\left((t)/(4) + 2\right) &= -2\left((t)/(4) + 2\right)^2 - 5 \\ \\ &= -2\left((t^2)/(16) +t + 4\right) - 5 \\ \\ &= -(t^2)/(8) -2t -8 -5 \\ \\ &= -(1)/(8)t^2 - 2t - 13 \end{aligned}`

In conclusion:

`\displaystyle \left( g\circ f\right)\left((t)/(4)\right) = -(1)/(8)t^2 - 2t - 13`