Question

1. selected values of an exponential function, f, are given in the table. complete the rest of the table.

x	0	1	2	3	4	5
f(x)		24	36

Explanation:

Step1: Find the common ratio

For an exponential function \( f(x) = ab^x \), the ratio between consecutive terms is constant (the base \( b \)). We know \( f(1) = 24 \) and \( f(2) = 36 \). So the common ratio \( b=\frac{f(2)}{f(1)}=\frac{36}{24}=\frac{3}{2} = 1.5\).

Step2: Find \( f(0) \)

For an exponential function \( f(x)=ab^x \), when \( x = 0 \), \( f(0)=ab^0=a\times1=a \). We can also find \( a \) using \( f(1)=ab^1 \). We know \( f(1) = 24 \) and \( b = 1.5 \), so \( 24=a\times1.5\), then \( a=\frac{24}{1.5}=16 \). So \( f(0) = 16 \).

Step3: Find \( f(3) \)

Using the exponential function formula \( f(x)=ab^x \), with \( a = 16 \), \( b = 1.5 \) and \( x = 3 \), we have \( f(3)=16\times(1.5)^3=16\times3.375 = 54 \).

Step4: Find \( f(4) \)

Using \( f(x)=ab^x \), with \( a = 16 \), \( b = 1.5 \) and \( x = 4 \), we get \( f(4)=16\times(1.5)^4=16\times5.0625 = 81 \).

Step5: Find \( f(5) \)

Using \( f(x)=ab^x \), with \( a = 16 \), \( b = 1.5 \) and \( x = 5 \), we have \( f(5)=16\times(1.5)^5=16\times7.59375 = 121.5 \).

Answer:

The completed table is:

\( x \)	0	1	2	3	4	5