how does $f(t) = 5^t$ change over the interval from $t = -1$ to $t = 4$?\n$f(t)$ increases by $5^5$\n$f(t)$…

how does $f(t) = 5^t$ change over the interval from $t = -1$ to $t = 4$?\n$f(t)$ increases by $5^5$\n$f(t)$ decreases by $5^5$\n$f(t)$ decreases by a factor of $5^6$\n$f(t)$ increases by a factor of $5^5$

how does $f(t) = 5^t$ change over the interval from $t = -1$ to $t = 4$?\n$f(t)$ increases by $5^5$\n$f(t)$ decreases by $5^5$\n$f(t)$ decreases by a factor of $5^6$\n$f(t)$ increases by a factor of $5^5$

Answer

Explanation:

Step1: Find f(-1)

Substitute ( t = -1 ) into ( f(t) = 5^t ), so ( f(-1)=5^{-1}=\frac{1}{5} ).

Step2: Find f(4)

Substitute ( t = 4 ) into ( f(t) = 5^t ), so ( f(4)=5^{4} ).

Step3: Determine the factor of change

To find how much ( f(t) ) changes from ( t=-1 ) to ( t = 4 ), we calculate the ratio ( \frac{f(4)}{f(-1)}=\frac{5^{4}}{5^{-1}} ). Using the rule of exponents ( \frac{a^m}{a^n}=a^{m - n} ), we get ( 5^{4-(-1)}=5^{5} ). Since ( 5^{5}>1 ), ( f(t) ) increases by a factor of ( 5^{5} ).

Answer:

f(t) increases by a factor of ( 5^5 ) (the last option: f(t) increases by a factor of ( 5^5 ))