The function f is defined by f(x) = a \cdot 3^x, where a is a constant. If f(2) = 45, what… — SAT

The function $f$ is defined by $f(x) = a \cdot 3^x$ , where $a$ is a constant. If $f(2) = 45$ , what is the value of $f(1)$ ?

Answer: 15

To find the value of

f(1)

, we first determine the value of the constant

a

. We are given that

f(2) = 45

, so substituting

x = 2

into the function definition

f(x) = a \cdot 3^x

gives

45 = a \cdot 3^2

. Simplifying

3^2

9

yields

45 = 9a

, which means

a = 5

. Now that we know

a = 5

, we can write the function as

f(x) = 5 \cdot 3^x

. To find

f(1)

, we substitute

x = 1

into this equation, yielding

f(1) = 5 \cdot 3^1 = 15

Substitute the given point

(2, 45)

into the function equation to solve for

a

a = 5

Since

f(2) = 45

, we have

45 = a \cdot 3^2 = 9a

, which gives

a = 5

Evaluate the function at

x = 1

using the value of

a = 5

f(1) = 15

Substituting

a = 5

and

x = 1

into

f(x) = a \cdot 3^x

gives

f(1) = 5 \cdot 3^1 = 15

Evaluating and solving exponential functions given initial conditions or points.

Question