Question

The table represents some points on the graph of an exponential function. Which function represents the relationship?

x f(x)
2 6
3 3
4 1.5
5 0.75
6 0.375

Answer 1

To represent the relationship of the given points with an exponential function, we can express the function in the form:

\[
f(x) = a \cdot b^{x}
\]

where \( a \) is the initial value when \( x \) is 0, and \( b \) is the base of the exponential function which determines the rate of growth or decay.

Let's first analyze the points provided:

- \( f(2) = 6 \)
- \( f(3) = 3 \)
- \( f(4) = 1.5 \)
- \( f(5) = 0.75 \)
- \( f(6) = 0.375 \)

From these points, we can notice that \( f(x) \) seems to be decreasing as \( x \) increases. This suggests that \( b \) is a fraction (since the function is decreasing).

Let's explore the relationship between the \( f(x) \) values:

1. From \( f(2) \) to \( f(3) \):
\[
\frac{f(3)}{f(2)} = \frac{3}{6} = \frac{1}{2}
\]

2. From \( f(3) \) to \( f(4) \):
\[
\frac{f(4)}{f(3)} = \frac{1.5}{3} = \frac{1}{2}
\]

3. From \( f(4) \) to \( f(5) \):
\[
\frac{f(5)}{f(4)} = \frac{0.75}{1.5} = \frac{1}{2}
\]

4. From \( f(5) \) to \( f(6) \):
\[
\frac{f(6)}{f(5)} = \frac{0.375}{0.75} = \frac{1}{2}
\]

Since the ratios of successive function values are consistent at \( \frac{1}{2} \), we can conclude that \( b = \frac{1}{2} \).

Now, we need to find \( a \). We can use any point, say \( f(2) = 6 \):

\[
f(2) = a \cdot b^2 = a \cdot \left(\frac{1}{2}\right)^2 = a \cdot \frac{1}{4}
\]
\[
6 = a \cdot \frac{1}{4}
\]
\[
a = 6 \cdot 4 = 24
\]

Thus we can now write the function:

\[
f(x) = 24 \cdot \left(\frac{1}{2}\right)^{x}
\]

This can also be expressed as:

\[
f(x) = 24 \cdot 2^{-x}
\]

So, the function representing the relationship is:

\[
f(x) = 24 \cdot 2^{-x}
\]