Question

this table shows several values of an exponential function.
x | f(x)
0 | 4
1 | 20
2 | 100
3 | 500
write an equation for the function in the form f(x) = a(b)^x.
use whole numbers, decimals, or simplified fractions for the values of a and b.
f(x) =

Explanation:

Step1: Find the value of \( a \)

For an exponential function \( f(x) = a(b)^x \), when \( x = 0 \), \( f(0)=a(b)^0=a(1)=a \). From the table, when \( x = 0 \), \( f(0) = 4 \), so \( a = 4 \).

Step2: Find the value of \( b \)

Now we know \( a = 4 \), so the function is \( f(x)=4(b)^x \). Let's use another point, say \( x = 1 \), \( f(1)=20 \). Substitute into the function: \( 20 = 4(b)^1 \). Solve for \( b \): divide both sides by 4, \( \frac{20}{4}=b \), so \( b = 5 \). We can check with \( x = 2 \): \( f(2)=4(5)^2=4\times25 = 100 \), which matches the table. And \( x = 3 \): \( f(3)=4(5)^3=4\times125 = 500 \), which also matches.

Answer:

\( f(x) = 4(5)^x \)