Question

question 2
let $f(t) = 5t + 3$ and $g(t) = 7^t$.
a) find $(f(g(t))$ and simplify the result.
$(f(g(t)) =

b) find $(g(f(t))$ and simplify the result.
$(g(f(t)) =

question help: video

Explanation:

Part (a)

Step1: Substitute \( g(t) \) into \( f(t) \)

We know \( f(t) = 5t + 3 \) and \( g(t) = 7^t \). To find \( f(g(t)) \), we replace \( t \) in \( f(t) \) with \( g(t) \). So we substitute \( 7^t \) for \( t \) in the function \( f(t) \).
\( f(g(t)) = f(7^t) = 5(7^t) + 3 \)

Step2: Simplify the expression

The expression \( 5(7^t) + 3 \) can be written as \( 5 \cdot 7^t + 3 \), and this is already in a simplified form.

Step1: Substitute \( f(t) \) into \( g(t) \)

We know \( g(t) = 7^t \) and \( f(t) = 5t + 3 \). To find \( g(f(t)) \), we replace \( t \) in \( g(t) \) with \( f(t) \). So we substitute \( 5t + 3 \) for \( t \) in the function \( g(t) \).
\( g(f(t)) = g(5t + 3) = 7^{(5t + 3)} \)

Step2: Simplify the expression (if possible)

The expression \( 7^{(5t + 3)} \) can be simplified using the exponent rule \( a^{m + n}=a^m\cdot a^n \). So \( 7^{5t + 3}=7^{5t}\cdot7^{3} \). And \( 7^{3}=343 \), so \( 7^{5t + 3}=343\cdot7^{5t} \) (or we can leave it as \( 7^{5t + 3} \) which is also a simplified form).

Answer:

\( f(g(t)) = 5 \cdot 7^t + 3 \) (or \( 5(7^t) + 3 \))

Part (b)