Question

for the function $f(x) = 5e^x$ and $g(x) = x^3$, find the following: (a) $f(g(1))=$ (b) $g(f(1))=$ (c) $f(g(x))=$ (d) $g(f(x))=$ (e) $f(t)g(t)=$

Explanation:

Step1: Compute g(1)

$g(1)=1^3=1$

Step2: Compute f(g(1))

$f(g(1))=f(1)=5e^1=5e$

Step3: Compute f(1)

$f(1)=5e^1=5e$

Step4: Compute g(f(1))

$g(f(1))=(5e)^3=125e^3$

Step5: Substitute g(x) into f

$f(g(x))=5e^{g(x)}=5e^{x^3}$

Step6: Substitute f(x) into g

$g(f(x))=(f(x))^3=(5e^x)^3=125e^{3x}$

Step7: Multiply f(t) and g(t)

$f(t)g(t)=5e^t \cdot t^3=5t^3e^t$

Answer:

(a) $5e$
(b) $125e^3$
(c) $5e^{x^3}$
(d) $125e^{3x}$
(e) $5t^3e^t$