For the function f, selected values of x and f(x) are shown in the table below. | x | f(x) | | :---: | :---: | | -1 | 4 | | 1 | 2 | | 3 | -1 | | 5 | 6 | The function g is defined by g(x) = a \cdot f(x - 2) + 5, where a is a constant. If g(5) = 3, what is the value of a?

For the function f, selected values of x and f(x) are shown in the table below. | x | f(x)… — SAT

For the function $f$ , selected values of $x$ and $f(x)$ are shown in the table below.

The function $g$ is defined by $g(x) = a \cdot f(x - 2) + 5$ , where $a$ is a constant. If $g(5) = 3$ , what is the value of $a$ ?

Answer: 2

Evaluating

g(5)

using the formula

g(x) = a \cdot f(x - 2) + 5

requires finding

f(5 - 2) = f(3)

. From the table,

f(3) = -1

. Substituting these values yields

3 = a(-1) + 5

, which simplifies to

-a = -2

, or

a = 2

Express

g(5)

using the given definition of

g(x)

g(5) = a \cdot f(3) + 5

By substituting

x = 5

into the definition

g(x) = a \cdot f(x - 2) + 5

, we obtain

g(5) = a \cdot f(5 - 2) + 5 = a \cdot f(3) + 5

Find the value of

f(3)

from the table.

f(3) = -1

Looking at the row where

x = 3

in the table, the corresponding output

f(x)

-1

Substitute the known values into the equation for

g(5)

and solve for

a

a = 2

Substitute

g(5) = 3

and

f(3) = -1

into the equation to get

3 = a(-1) + 5

. Subtracting 5 from both sides gives

-2 = -a

, which simplifies to

a = 2

Evaluating a transformed function using a table of values.