Question

1. $f(x) = \frac{3(x + 2)}{(x - 2)(x + 2)}$

x - intercept:
v.a.
hole:
domain:
y - intercept:
h.a.
range:

1. $f(x) = \frac{-3(x - 1)}{x(x - 1)}$

x - intercept:
v.a.
hole:
domain:
y - intercept:
h.a.
range:
graphing rational functions with holes

Explanation:

For Function 1: $f(x)=\frac{3(x+2)}{(x-2)(x+2)}$

Step1: Simplify the function

Cancel common factor $(x+2)$:
$f(x)=\frac{3}{x-2}$, where $x
eq-2$

Step2: Find x-intercept

Set $f(x)=0$:
$0=\frac{3}{x-2}$ → No solution, so no x-intercept.

Step3: Verify y-intercept

Set $x=0$:
$f(0)=\frac{3}{0-2}=-\frac{3}{2}=-1.5$

Step4: Find Vertical Asymptote (V.A.)

Set denominator of simplified function to 0:
$x-2=0$ → $x=2$

Step5: Find Horizontal Asymptote (H.A.)

Degree of numerator < degree of denominator:
$y=0$

Step6: Find hole

Canceled factor gives hole at $x=-2$. Find y-value:
$f(-2)=\frac{3}{-2-2}=-\frac{3}{4}=-0.75$

Step7: Determine domain

Exclude values making denominator 0:
$x
eq2, x
eq-2$

Step8: Determine range

Exclude horizontal asymptote value, plus hole y-value:
$y
eq0, y
eq-0.75$

Step1: Simplify the function

Cancel common factor $(x-1)$:
$f(x)=-\frac{3}{x}$, where $x
eq1$

Step2: Find x-intercept

Set $f(x)=0$:
$0=-\frac{3}{x}$ → No solution, so no x-intercept.

Step3: Find y-intercept

Set $x=0$:
Denominator is 0, so no y-intercept.

Step4: Find Vertical Asymptote (V.A.)

Set denominator of simplified function to 0:
$x=0$

Step5: Find Horizontal Asymptote (H.A.)

Degree of numerator < degree of denominator:
$y=0$

Step6: Find hole

Canceled factor gives hole at $x=1$. Find y-value:
$f(1)=-\frac{3}{1}=-3$

Step7: Determine domain

Exclude values making denominator 0:
$x
eq0, x
eq1$

Step8: Determine range

Exclude horizontal asymptote value, plus hole y-value:
$y
eq0, y
eq-3$

Answer:

for Function 1:
x-intercept: None
y-intercept: $(0, -1.5)$
V.A.: $x=2$
H.A.: $y=0$
Hole: $(-2, -0.75)$
Domain: $x
eq -2, 2$
Range: $y
eq 0, -0.75$

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For Function 2: $f(x)=\frac{-3(x-1)}{x(x-1)}$