Question

the table shows some data for an exponential function.
x: -2, -1, 0, 1, 2
P(x): 2,500, 500, 100, 20, 4
which function represents the relation shown in the table?
a P(x) = 100(1/5)^x
b P(x) = 20(1/5)^x
c P(x) = 100(5)^x
d P(x) = 20(5)^x

Answer 1

To determine which function represents the relation shown in the table, we need to analyze the given data points:

- When \( x = -2 \), \( P(-2) = 2500 \)
- When \( x = -1 \), \( P(-1) = 500 \)
- When \( x = 0 \), \( P(0) = 100 \)
- When \( x = 1 \), \( P(1) = 20 \)
- When \( x = 2 \), \( P(2) = 4 \)

Given the choices:
a) \( P(x) = 100(1/5)^x \)
b) \( P(x) = 20(1/5)^x \)
c) \( P(x) = 100(5)^x \)
d) \( P(x) = 20(5)^x \)

Let's evaluate each function at \( x = 0 \) to find the constant:

1. **Evaluate Option a**:
\[
P(0) = 100(1/5)^0 = 100
\]

2. **Evaluate Option b**:
\[
P(0) = 20(1/5)^0 = 20
\]

3. **Evaluate Option c**:
\[
P(0) = 100(5)^0 = 100
\]

4. **Evaluate Option d**:
\[
P(0) = 20(5)^0 = 20
\]

Since we see that the function must equal \( P(0) = 100 \), both options a and c seem possible. Next, let's check the values of the other \( x \) values with these functions (a and c):

**Testing Option a: \( P(x) = 100(1/5)^x \)**:
- For \( x = -2 \):
\[
P(-2) = 100(1/5)^{-2} = 100 \cdot 25 = 2500
\]
- For \( x = -1 \):
\[
P(-1) = 100(1/5)^{-1} = 100 \cdot 5 = 500
\]
- For \( x = 1 \):
\[
P(1) = 100(1/5)^1 = 100 \cdot \frac{1}{5} = 20
\]
- For \( x = 2 \):
\[
P(2) = 100(1/5)^2 = 100 \cdot \frac{1}{25} = 4
\]

**Testing Option c: \( P(x) = 100(5)^x \)**:
- For \( x = -2 \):
\[
P(-2) = 100(5)^{-2} = 100 \cdot \frac{1}{25} = 4 \quad \text{(not correct)}
\]
- This shows Option c does not match the table.

Therefore, the correct function that represents the relation is:
**a) \( P(x) = 100(1/5)^x \)**.