Question

math college liberal arts
name________
functions introduction
date______ period______
example 1 given the equation $y = x + 3$,
we can write the function $f(x) = x + 3$
find the following $f(x)$ values:
$f(-4) =$
$f(-2) =$
$f(0) =$
$f(2) =$
$f(4) =$
domain:
range:
increasing, decreasing, or constant?
example 2 given the equation $y = -\frac{1}{2}x - 5$,
we can write the function $f(x) = -\frac{1}{2}x - 5$
find the following $f(x)$ values:
$f(-8) =$
$f(-4) =$
$f(0) =$
$f(4) =$
$f(8) =$
domain:
range:
increasing, decreasing, or constant?

Explanation:

Example 1: $f(x)=x+3$

Step1: Substitute $x=-4$

$f(-4) = -4 + 3$

Step2: Calculate result

$f(-4) = -1$

Step3: Substitute $x=-2$

$f(-2) = -2 + 3$

Step4: Calculate result

$f(-2) = 1$

Step5: Substitute $x=0$

$f(0) = 0 + 3$

Step6: Calculate result

$f(0) = 3$

Step7: Substitute $x=2$

$f(2) = 2 + 3$

Step8: Calculate result

$f(2) = 5$

Step9: Substitute $x=4$

$f(4) = 4 + 3$

Step10: Calculate result

$f(4) = 7$

Step11: Identify domain

Linear functions have no restrictions, so domain is all real numbers.

Step12: Identify range

Linear functions output all real numbers, so range is all real numbers.

Step13: Analyze slope sign

Slope $m=1>0$, so function is increasing.

Example 2: $f(x)=-\frac{1}{2}x-5$

Step1: Substitute $x=-8$

$f(-8) = -\frac{1}{2}(-8) - 5$

Step2: Calculate result

$f(-8) = 4 - 5 = -1$

Step3: Substitute $x=-4$

$f(-4) = -\frac{1}{2}(-4) - 5$

Step4: Calculate result

$f(-4) = 2 - 5 = -3$

Step5: Substitute $x=0$

$f(0) = -\frac{1}{2}(0) - 5$

Step6: Calculate result

$f(0) = 0 - 5 = -5$

Step7: Substitute $x=4$

$f(4) = -\frac{1}{2}(4) - 5$

Step8: Calculate result

$f(4) = -2 - 5 = -7$

Step9: Substitute $x=8$

$f(8) = -\frac{1}{2}(8) - 5$

Step10: Calculate result

$f(8) = -4 - 5 = -9$

Step11: Identify domain

Linear functions have no restrictions, so domain is all real numbers.

Step12: Identify range

Linear functions output all real numbers, so range is all real numbers.

Step13: Analyze slope sign

Slope $m=-\frac{1}{2}<0$, so function is decreasing.

Answer:

Example 1: $f(x)=x+3$

$f(-4) = -1$
$f(-2) = 1$
$f(0) = 3$
$f(2) = 5$
$f(4) = 7$
Domain: All real numbers ($(-\infty, \infty)$)
Range: All real numbers ($(-\infty, \infty)$)
Increasing, Decreasing, or Constant? Increasing

Example 2: $f(x)=-\frac{1}{2}x-5$

$f(-8) = -1$
$f(-4) = -3$
$f(0) = -5$
$f(4) = -7$
$f(8) = -9$
Domain: All real numbers ($(-\infty, \infty)$)
Range: All real numbers ($(-\infty, \infty)$)
Increasing, Decreasing, or Constant? Decreasing