Question

based on the function notation provided, evaluate the function to determine the output.
example:
instead of using x and y in an equation, like y = 5x + 1, mathematicians often write f(n)=5n + 1 because it can give more information. with this notation, the direction to find f(2) means to replace the value of n with 2 and perform the operations to find f(n). the point (n, f(n)) is the same location on the graph as (x,y), where n describes the location along the x - axis, and f(n) is the height of the graph.
given that f(n)=4n + 1 and g(n)=7n - 3, evaluate the following functions with the indicated values.

1. f(1)=
2. g(0)=
3. f(5)=
4. g(-4)=

write each expression using an exponent.

1. 5×5×5×5
2. 2×2×2×2×2×2

10.\frac{1}{2}×\frac{1}{2}×\frac{1}{2}
11.\frac{2}{3}×\frac{2}{3}
write each expression in expanded form and then calculate the value of the expression.
12.4^1
13.6^3
14.5(2)^4

Explanation:

Step1: Evaluate f(1)

Substitute n = 1 into f(n)=4n + 1.
$f(1)=4\times1 + 1=4 + 1=5$

Step2: Evaluate g(0)

Substitute n = 0 into g(n)=7n - 3.
$g(0)=7\times0-3=0 - 3=-3$

Step3: Evaluate f(5)

Substitute n = 5 into f(n)=4n + 1.
$f(5)=4\times5+1=20 + 1=21$

Step4: Evaluate g(-4)

Substitute n=-4 into g(n)=7n - 3.
$g(-4)=7\times(-4)-3=-28-3=-31$

Step5: Write 5×5×5×5 in exponential form

According to the rule $a\times a\times\cdots\times a=a^n$ (n times of a), $5\times5\times5\times5 = 5^4$

Step6: Write $\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}$ in exponential form

$\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}=(\frac{1}{2})^3$

Step7: Write 2×2×2×2×2×2 in exponential form

$2\times2\times2\times2\times2\times2 = 2^6$

Step8: Write $\frac{2}{3}\times\frac{2}{3}$ in exponential form

$\frac{2}{3}\times\frac{2}{3}=(\frac{2}{3})^2$

Step9: Write $4^1$ in expanded form and calculate

Expanded form is 4, and its value is 4.

Step10: Write $6^3$ in expanded form and calculate

Expanded form is $6\times6\times6$, and $6\times6\times6 = 36\times6=216$

Step11: Write $5(2)^4$ in expanded form and calculate

First, $(2)^4=2\times2\times2\times2 = 16$, then $5(2)^4=5\times16 = 80$

Answer:

1. $f(1)=5$
2. $g(0)=-3$
3. $f(5)=21$
4. $g(-4)=-31$
5. $5^4$
6. $2^6$
7. $(\frac{1}{2})^3$
8. $(\frac{2}{3})^2$
9. Expanded form: 4, Value: 4
10. Expanded form: $6\times6\times6$, Value: 216
11. Expanded form: $5\times(2\times2\times2\times2)$, Value: 80