For a linear function f, the equation f(x + 3) - f(x - 1) = 16 is true for all real number… — SAT

For a linear function $f$ , the equation $f(x + 3) - f(x - 1) = 16$ is true for all real numbers $x$ . If the graph of $y = f(x)$ in the $xy$ -plane passes through the point $(2, 5)$ , what is the $x$ -intercept of the graph of $f$ ?

A
$-\frac{3}{4}$
$\frac{3}{4}$ Answer
C
$\frac{11}{8}$
D
$-18$

Answer

The

x

-intercept of the graph of

f

\frac{3}{4}

By writing the linear function as

f(x) = mx + b

, we can evaluate

f(x+3) - f(x-1)

(mx + 3m + b) - (mx - m + b) = 4m

. Setting this equal to the given value of

16

yields a slope of

m = 4

. Using the point

(2, 5)

f(2) = 4(2) + b = 5

gives the

y

-intercept

b = -3

, so the function is

f(x) = 4x - 3

. The

x

-intercept is found by setting

f(x) = 0

, which yields

x = \frac{3}{4}

Step-by-Step Solution

Represent the linear function in slope-intercept form

f(x) = mx + b

and substitute the expressions

x+3

and

x-1

into the function.

f(x+3) = m(x+3) + b

and

f(x-1) = m(x-1) + b

This allows us to evaluate the given functional equation using the general parameters of a linear function.

Calculate the difference

f(x+3) - f(x-1)

and set it equal to

16

to find the slope

m

(mx + 3m + b) - (mx - m + b) = 4m = 16

, which simplifies to

m = 4

The difference between two values of a linear function is directly proportional to the difference in their inputs, where the constant of proportionality is the slope.

Substitute the point

(2, 5)

and the slope

m = 4

into

f(x) = 4x + b

to find the

y

-intercept

b

5 = 4(2) + b \implies 5 = 8 + b \implies b = -3

A linear function is completely defined once both its slope and a point on its graph are known.

Find the

x

-intercept of the graph by setting

f(x) = 0

and solving for

x

4x - 3 = 0 \implies x = \frac{3}{4}

The

x

-intercept of a function's graph is the input value at which the function's output equals zero.

Key Concept

Determining the slope, equation, and intercepts of a linear function from functional relationships and coordinate points.

Estimated Time:2m 0s

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