Question

function notation
when a function can be written as an equation, the symbol f(x) replaces y and is read as \the value of f at x\ or simply \f of x.\
this does not mean f times x.
replacing y with f(x) is called writing a function in function notation.
examples:
if f(x) = 2x − 3, find the following:
a. f(-2) b. f(7) c. f(-4)
if k(x) = -7x + 1, find the following:
d. k(0) e. k(-1) f. k(5)
sometimes, there will be multiple x’s in an equation. when this occurs, simply replace both values of x.
if h(x) = x² − 3x + 5, find the following:
a. h(-3) b. h(5)
if p(x) = x² + 5x − 3, find the following:
c. p(-2) d. p(1)
if f(x) = 5x - 3, fill out the following table of values:

x	-2	-1	0	1	2	3
f(x)

what type of function is this?
remember***
f(-3) means -3 is your input and you plug it in for x
f(x) = -3 means that your whole function is = to -3 and you plug into the y.

Explanation:

Let's solve each part step by step. We'll start with the first set of functions:

Part 1: \( f(x) = 2x - 3 \)

a. \( f(-2) \)

Step1: Substitute \( x = -2 \) into \( f(x) \)

\( f(-2) = 2(-2) - 3 \)

Step2: Calculate

\( 2(-2) = -4 \), so \( -4 - 3 = -7 \)

b. \( f(7) \)

Step1: Substitute \( x = 7 \) into \( f(x) \)

\( f(7) = 2(7) - 3 \)

Step2: Calculate

\( 2(7) = 14 \), so \( 14 - 3 = 11 \)

c. \( f(-4) \)

Step1: Substitute \( x = -4 \) into \( f(x) \)

\( f(-4) = 2(-4) - 3 \)

Step2: Calculate

\( 2(-4) = -8 \), so \( -8 - 3 = -11 \)

Part 2: \( k(x) = -7x + 1 \)

d. \( k(0) \)

Step1: Substitute \( x = 0 \) into \( k(x) \)

\( k(0) = -7(0) + 1 \)

Step2: Calculate

\( -7(0) = 0 \), so \( 0 + 1 = 1 \)

e. \( k(-1) \)

Step1: Substitute \( x = -1 \) into \( k(x) \)

\( k(-1) = -7(-1) + 1 \)

Step2: Calculate

\( -7(-1) = 7 \), so \( 7 + 1 = 8 \)

f. \( k(5) \)

Step1: Substitute \( x = 5 \) into \( k(x) \)

\( k(5) = -7(5) + 1 \)

Step2: Calculate

\( -7(5) = -35 \), so \( -35 + 1 = -34 \)

Part 3: \( h(x) = x^2 - 3x + 5 \)

a. \( h(-3) \)

Step1: Substitute \( x = -3 \) into \( h(x) \)

\( h(-3) = (-3)^2 - 3(-3) + 5 \)

Step2: Calculate each term

\( (-3)^2 = 9 \), \( -3(-3) = 9 \), so \( 9 + 9 + 5 = 23 \)

b. \( h(5) \)

Step1: Substitute \( x = 5 \) into \( h(x) \)

\( h(5) = (5)^2 - 3(5) + 5 \)

Step2: Calculate each term

\( 5^2 = 25 \), \( -3(5) = -15 \), so \( 25 - 15 + 5 = 15 \)

Part 4: \( p(x) = x^2 + 5x - 3 \)

c. \( p(-2) \)

Step1: Substitute \( x = -2 \) into \( p(x) \)

\( p(-2) = (-2)^2 + 5(-2) - 3 \)

Step2: Calculate each term

\( (-2)^2 = 4 \), \( 5(-2) = -10 \), so \( 4 - 10 - 3 = -9 \)

d. \( p(1) \)

Step1: Substitute \( x = 1 \) into \( p(x) \)

\( p(1) = (1)^2 + 5(1) - 3 \)

Step2: Calculate each term

\( 1^2 = 1 \), \( 5(1) = 5 \), so \( 1 + 5 - 3 = 3 \)

Part 5: \( f(x) = 5x - 3 \) - Table of Values

For each \( x \), substitute into \( f(x) \):

• \( x = -2 \): \( f(-2) = 5(-2) - 3 = -10 - 3 = -13 \)
• \( x = -1 \): \( f(-1) = 5(-1) - 3 = -5 - 3 = -8 \)
• \( x = 0 \): \( f(0) = 5(0) - 3 = 0 - 3 = -3 \)
• \( x = 1 \): \( f(1) = 5(1) - 3 = 5 - 3 = 2 \)
• \( x = 2 \): \( f(2) = 5(2) - 3 = 10 - 3 = 7 \)
• \( x = 3 \): \( f(3) = 5(3) - 3 = 15 - 3 = 12 \)

The table becomes:

\( x \)	-2	-1	0	1	2	3

Type of Function:

\( f(x) = 5x - 3 \) is a linear function because it is in the form \( f(x) = mx + b \) where \( m = 5 \) (slope) and \( b = -3 \) (y-intercept).

Answer:

s:

• \( f(-2) = \boldsymbol{-7} \)
• \( f(7) = \boldsymbol{11} \)
• \( f(-4) = \boldsymbol{-11} \)
• \( k(0) = \boldsymbol{1} \)
• \( k(-1) = \boldsymbol{8} \)
• \( k(5) = \boldsymbol{-34} \)
• \( h(-3) = \boldsymbol{23} \)
• \( h(5) = \boldsymbol{15} \)
• \( p(-2) = \boldsymbol{-9} \)
• \( p(1) = \boldsymbol{3} \)
• Table values: \( -13, -8, -3, 2, 7, 12 \) (for \( x = -2, -1, 0, 1, 2, 3 \) respectively)
• Type of function: \(\boldsymbol{\text{Linear Function}}\)