Question

determine if the function below is linear or exponential then fill in the guided sentences below.

x	f(x)
2	9
3	27
4	81

answer
attempt 1 out of 2
the function is
. the
of this function is
exactly
.

Explanation:

Step1: Check linearity (constant difference)

For a linear function, the difference between consecutive \( f(x) \) values (when \( x \) increases by 1) should be constant.

• From \( x = 1 \) to \( x = 2 \): \( 9 - 3 = 6 \)
• From \( x = 2 \) to \( x = 3 \): \( 27 - 9 = 18 \)
• From \( x = 3 \) to \( x = 4 \): \( 81 - 27 = 54 \)

The differences (6, 18, 54) are not constant, so it's not linear.

Step2: Check exponentiality (constant ratio)

For an exponential function, the ratio of consecutive \( f(x) \) values (when \( x \) increases by 1) should be constant.

• From \( x = 1 \) to \( x = 2 \): \( \frac{9}{3} = 3 \)
• From \( x = 2 \) to \( x = 3 \): \( \frac{27}{9} = 3 \)
• From \( x = 3 \) to \( x = 4 \): \( \frac{81}{27} = 3 \)

The ratio is constant (3), so the function is exponential. The base of the exponential function (the constant ratio) is 3.

Answer:

The function is exponential because as \( x \) increases, the \( y \)-values change by a constant ratio of 3. The base of this function is exactly 3.