Question

(x)=(x+2)^2(x-1)^2." align="absmiddle" class="latex-formula">

Find the range

Answer 1

X
∈
R
,
x
≠
±
2

y
∈
R
,
y
≠
1

Explanation:
The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be.
solve
x
2
−
4
=
0
⇒
x
2
=
4

⇒
x
=
±
2
←
excluded values

⇒
domain is
x
∈
R
,
x
≠
±
2

To find the value that y cannot be find
lim
x
→
±
∞

f
(
x
)

divide terms on numerator/denominator by the highest power of x, that is
x
2

f
(
x
)
=
x
2
x
2
−
2
x
2
x
2
x
2
−
4
x
2
=
1
−
2
x
2
1
−
4
x
2

as
x
→
±
∞
,
f
(
x
)
→
1
−
0
1
−
0

⇒
y
→
1
←
excluded value

⇒
range is
y
∈
R
,
y
≠
1

The graph of f(x) illustrates this.
graph{(x^2-2)/(x^2-4) [-10, 10, -5, 5]}