Answer:
[tex]\displaystyle \left( g\circ f\right)\left(\frac{t}{4}\right) = -\frac{1}{8}t^2 - 2t - 13[/tex]
Step-by-step explanation:
We are given the two functions:
[tex]\displaystyle g(t) = -2t^2 -5 \text{ and } f(t) = t + 2[/tex]
And we want to find:
[tex]\displaystyle ( g \circ f)\left(\frac{t}{4}\right)[/tex]
Recall that this is equivalent to:
[tex]\displaystyle = g\left(f\left(\frac{t}{4}\right)\right)[/tex]
Find f(t / 4):
[tex]\displaystyle \begin{aligned} f\left(\frac{t}{4}\right) &= \left(\frac{t}{4}\right) + 2 \\ \\ &= \frac{t}{4} + 2\end{aligned}[/tex]
By substitution:
[tex]\displaystyle g\left(f\left(\frac{t}{4}\right)\right)= g\left(\frac{t}{4} +2\right)[/tex]
Find the above function:
[tex]\displaystyle \begin{aligned} g\left(\frac{t}{4} + 2\right) &= -2\left(\frac{t}{4} + 2\right)^2 - 5 \\ \\ &= -2\left(\frac{t^2}{16} +t + 4\right) - 5 \\ \\ &= -\frac{t^2}{8} -2t -8 -5 \\ \\ &= -\frac{1}{8}t^2 - 2t - 13 \end{aligned}[/tex]
In conclusion:
[tex]\displaystyle \left( g\circ f\right)\left(\frac{t}{4}\right) = -\frac{1}{8}t^2 - 2t - 13[/tex]
