In the xy-plane, the graph of the linear function f passes through the points (3, 11) and … — SAT

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(3, 11)$ and $(7, 23)$ . The function $g$ is defined by $g(x) = f(2x) - 5$ . What is the value of $g(4)$ ?

Answer: 21

Answer

The correct answer is 21. First, find the slope of the linear function

f

using the given points

(3, 11)

and

(7, 23)

m = \frac{23 - 11}{7 - 3} = 3

. The equation of the line is

f(x) - 11 = 3(x - 3)

, which simplifies to

f(x) = 3x + 2

. To find

g(4)

, substitute

x = 4

into the definition of

g(x)

g(4) = f(2 \cdot 4) - 5 = f(8) - 5

. Evaluating

f(8)

gives

3(8) + 2 = 26

. Finally, subtracting

5

gives

g(4) = 26 - 5 = 21

Step-by-Step Solution

Determine the equation of the linear function

f(x)

f(x) = 3x + 2

First find the slope

m = \frac{23 - 11}{7 - 3} = 3

. Then, use the point-slope formula with

(3, 11)

to get

f(x) - 11 = 3(x - 3)

, which simplifies to

f(x) = 3x + 2

Express

g(4)

in terms of

f

g(4) = f(8) - 5

Substitute

x = 4

into the definition

g(x) = f(2x) - 5

to get

g(4) = f(2(4)) - 5

Calculate the value of

f(8)

and

g(4)

g(4) = 21

Evaluate

f(8) = 3(8) + 2 = 26

, then subtract

5

to obtain

g(4) = 26 - 5 = 21

Key Concept

Linear Functions and Graphs

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic