Question

A 2-column table has 5 rows. The first column is labeled x with entries negative 2, negative 1, 0, 1, 2. The second column is labeled f (x) with entries 12. 5, 2. 5, 0. 5, 0. 1, 0. 2. Which exponential function is represented by the table? f(x) = 0. 2(0. 5x) f(x) = 0. 5(5x) f(x) = 0. 5(0. 2x) f(x) = 0. 2(0. 2x).

Answer 1

An exponential function is characterized by an initial value, and an exponential rate

The function f(x) is $\mathbf{f(x) =0.5(0.2)^x}$

f(x) is an exponential function that passes through points (-2,12.5), (-1,2.5), (0,0.5), (1,0.1) and (2,0.2).

An exponential function is represented as:

$\mathbf{y = ab^x}$

At point (0,0.5), we have:

$\mathbf{0.5 = ab^0}$

This gives

$\mathbf{0.5 = a \times 1}$

$\mathbf{0.5 = a}$

Rewrite as:

$\mathbf{a = 0.5 }$

At point (-1,2.5), we have:

$\mathbf{2.5 = ab^{-1}}$

Substitute 0.5 for a

$\mathbf{2.5 = 0.5b^{-1}}$

Divide both sides by 0.5

$\mathbf{5 = b^{-1}}$

Take inverse of both sides

$\mathbf{\frac 1{5} = b}$

Rewrite as:

$\mathbf{b = \frac 1{5} }$

$\mathbf{b = 0.2}$

So, we have:

$\mathbf{b = 0.2}$ and $\mathbf{a = 0.5 }$

Substitute these values in $\mathbf{y = ab^x}$

$\mathbf{y =0.5(0.2)^x}$

Express as a function

$\mathbf{f(x) =0.5(0.2)^x}$

Hence, the function f(x) is $\mathbf{f(x) =0.5(0.2)^x}$

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