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Rational Functions
Related Topics
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Rational Functions
6 Activities
Solve the following inequality for
x
x
x
x
−
1
x
+
1
≤
x
+
2
x
−
1
\frac{x-1}{x+1} \leq \frac{x+2}{x-1}
x
+
1
x
−
1
≤
x
−
1
x
+
2
x
≤
−
1
,
−
1
5
<
x
≤
1
x \leq -1, -\frac{1}{5} < x \leq 1
x
≤
−
1
,
−
5
1
<
x
≤
1
x
>
−
2
x>-2
x
>
−
2
−
1
<
x
≤
−
1
5
,
x
>
1
-1 < x \leq -\frac{1}{5}, x>1
−
1
<
x
≤
−
5
1
,
x
>
1
I don't know
Check Submission
More Rational Functions Questions:
Rational Functions: Slant Asymptote
Find the slant asymptote to the curve
y
=
x
3
+
x
2
+
1
x
2
+
1
\displaystyle y=\frac{x^3+x^2+1}{x^2+1}
y
=
x
2
+
1
x
3
+
x
2
+
1
.
Rational Functions
Q
:
\bf{Q:}
Q
:
Find the exact list of vertical asymptotes of
f
(
x
)
=
x
−
1
x
2
+
x
\displaystyle f\left(x\right)=\frac{x-1}{x^2+x}
f
(
x
)
=
x
2
+
x
x
−
1
Rational Functions: Slant Asymptote
Find the slant asymptote to the curve
y
=
x
3
+
x
2
+
1
x
2
+
1
\displaystyle y=\frac{x^3+x^2+1}{x^2+1}
y
=
x
2
+
1
x
3
+
x
2
+
1
.
Q
:
\bf{Q:}
Q
:
Find
f
(
g
(
x
)
)
f(g(x))
f
(
g
(
x
))
and its domain if
f
(
x
)
=
1
x
−
1
f\left(x\right)=\dfrac{1}{x-1}
f
(
x
)
=
x
−
1
1
and
g
(
x
)
=
1
x
+
2
g\left(x\right)=\dfrac{1}{x+2}
g
(
x
)
=
x
+
2
1
Solve the following inequality:
2
x
−
1
x
−
5
≥
x
+
1
x
+
5
\dfrac{2x-1}{x-5}\ge\dfrac{x+1}{x+5}
x
−
5
2
x
−
1
≥
x
+
5
x
+
1
Rational Functions: Inequalities with Fractions
Find all
x
x
x
that satisfy
x
+
3
x
2
+
8
x
+
15
≤
0
\dfrac{x+3}{x^2+8x+15}\le0
x
2
+
8
x
+
15
x
+
3
≤
0
Rational Functions
Find all
x
x
x
that satisfy
x
x
−
3
≥
4
\dfrac{x}{x-3}\ge4
x
−
3
x
≥
4
Rules of Exponents
Simplify:
−
5
x
2
y
a
3
3
z
5
⋅
3
7
x
b
y
15
z
−
2
-\dfrac{5x^2y^a}{3^3z^5}\cdot\dfrac{3^7x^by}{15z^{-2}}
−
3
3
z
5
5
x
2
y
a
⋅
15
z
−
2
3
7
x
b
y
Rational Functions
Let
f
(
x
)
=
x
2
+
5
x
+
6
x
−
4
f(x)=\dfrac{x^2+5x+6}{x-4}
f
(
x
)
=
x
−
4
x
2
+
5
x
+
6
be a rational function.
Solving Rational Equations
Practice: Solving Rational Equations
Let
f
(
x
)
=
4
x
2
−
25
2
x
+
5
f(x)=\dfrac{4x^2-25}{2x+5}
f
(
x
)
=
2
x
+
5
4
x
2
−
25
.
If
f
(
x
)
=
4
x
2
−
2
x
+
1
f(x)=4x^2-2x+1
f
(
x
)
=
4
x
2
−
2
x
+
1
, then what is
x
?
x?
x
?
Solving Rational Equations
Practice: Solving Rational Equations
Let
f
(
x
)
=
x
2
−
4
x
−
2
f(x)=\dfrac{x^2-4}{x-2}
f
(
x
)
=
x
−
2
x
2
−
4
.
If
f
(
x
)
=
x
2
−
5
x
−
6
f(x)=x^2-5x-6
f
(
x
)
=
x
2
−
5
x
−
6
, then what is
x
?
x?
x
?
Solving Rational Equations
Practice: Solving Rational Equations
Given the equation
4
x
x
−
5
+
3
x
+
2
=
x
x
2
−
3
x
−
10
\dfrac{4x}{x-5}+\dfrac{3}{x+2}=\dfrac{x}{x^2-3x-10}
x
−
5
4
x
+
x
+
2
3
=
x
2
−
3
x
−
10
x
, answer the following questions.
Solving Rational Equations
Practice: Solving Rational Equations
Given the equation
4
x
+
1
−
1
x
+
3
=
x
+
2
x
2
+
4
x
+
3
\dfrac{4}{x+1}-\dfrac{1}{x+3}=\dfrac{x+2}{x^2+4x+3}
x
+
1
4
−
x
+
3
1
=
x
2
+
4
x
+
3
x
+
2
, answer the following questions.
Solving Rational Equations
Practice: Solving Rational Equations
True or false:
81
26
\dfrac{81}{26}
26
81
is a solution to
10
=
4
x
−
1
3
x
+
8
10=\dfrac{4x-1}{3x+8}
10
=
3
x
+
8
4
x
−
1
.
Solving Rational Equations
Practice: Solving Rational Equations
True or false:
17
7
\dfrac{17}{7}
7
17
is a solution to
−
5
=
2
x
+
3
x
−
4
-5=\dfrac{2x+3}{x-4}
−
5
=
x
−
4
2
x
+
3
.
Solving Rational Equations
Solve for
x
x
x
:
5
2
x
−
1
−
7
3
x
+
2
=
9
6
x
2
+
x
−
2
\dfrac{5}{2x-1}-\dfrac{7}{3x+2}=\dfrac{9}{6x^2+x-2}
2
x
−
1
5
−
3
x
+
2
7
=
6
x
2
+
x
−
2
9
Solving Rational Equations
Solve
2
x
−
3
x
2
−
3
=
5
x
+
1
\dfrac{2x-3}{x^2-3}=\dfrac{5}{x+1}
x
2
−
3
2
x
−
3
=
x
+
1
5
.
Graphing Rational Functions
Practice: Graphing Rational Functions
Write a rational function with the following properties:
An oblique asymptote at
y
=
−
3
x
+
1
y=-3x+1
y
=
−
3
x
+
1
Graphing Rational Functions
Practice: Graphing Rational Functions
Write a rational function with the following properties:
An oblique asymptote at
y
=
2
x
−
7
y=2x-7
y
=
2
x
−
7
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form
f
(
x
)
=
a
x
+
b
c
x
+
d
f(x)=\displaystyle\frac{ax+b}{cx+d}
f
(
x
)
=
c
x
+
d
a
x
+
b
that describes the following:
Vertical asymptote at
x
=
−
1
x=-1
x
=
−
1
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form
f
(
x
)
=
a
x
+
b
c
x
+
d
f(x)=\displaystyle\frac{ax+b}{cx+d}
f
(
x
)
=
c
x
+
d
a
x
+
b
that describes the following:
Vertical asymptote at
x
=
−
2
x=-2
x
=
−
2
Graphing Rational Functions
Practice: Graphing Rational Functions
Sketch a graph of
f
(
x
)
=
x
2
−
25
x
+
5
f(x)=\dfrac{x^2-25}{x+5}
f
(
x
)
=
x
+
5
x
2
−
25
, finding all asymptotes, x-intercepts, missing points, and positive & negative intervals.
Graphing Rational Functions
Practice: Graphing Rational Functions
Sketch a graph of
f
(
x
)
=
−
x
+
9
x
+
3
f(x)=\dfrac{-x+9}{x+3}
f
(
x
)
=
x
+
3
−
x
+
9
, finding all asymptotes, intercepts, missing points, and positive & negative intervals.
Graphing Rational Functions
Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at
y
=
0
y=0
y
=
0
.
Graphing Rational Functions
Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at
y
=
0
y=0
y
=
0
.
Rational Functions
Practice: Rational Functions
Match the equation for the rational function with its appropriate properties.
Rational Functions
Practice: Rational Functions
Match the equation for the rational function with its appropriate properties.
Rational Functions
Practice: Rational Functions
Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of:
f
(
x
)
=
3
x
2
+
14
x
+
8
x
2
−
2
x
−
14
f(x)=\dfrac{3x^2+14x+8}{x^2-2x-14}
f
(
x
)
=
x
2
−
2
x
−
14
3
x
2
+
14
x
+
8
Rational Functions
Practice: Rational Functions
Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of:
f
(
x
)
=
2
x
2
+
x
−
1
2
x
2
+
5
x
−
3
f(x)=\dfrac{2x^2+x-1}{2x^2+5x-3}
f
(
x
)
=
2
x
2
+
5
x
−
3
2
x
2
+
x
−
1
Rational Functions
Practice: Rational Functions
Match the rational function with the correct sketch of its graph.
Rational Functions
Practice: Rational Functions
Match the rational function with the correct sketch of its graph.
Rational Functions
Which of the following graphs is a good approximation of
f
(
x
)
=
x
2
−
3
x
x
+
6
f(x) = \frac{x^2 - 3x}{x + 6}
f
(
x
)
=
x
+
6
x
2
−
3
x
for
x
x
x
on the small interval
[
−
1
/
10
,
1
/
10
]
[-1/10, 1/10]
[
−
1/10
,
1/10
]
Rational Functions
Match the following five functions with the above graphs.
Rules of Exponents
Simplify:
−
5
x
2
y
a
3
3
z
5
⋅
3
7
x
b
y
15
z
−
2
-\dfrac{5x^2y^a}{3^3z^5}\cdot\dfrac{3^7x^by}{15z^{-2}}
−
3
3
z
5
5
x
2
y
a
⋅
15
z
−
2
3
7
x
b
y
Rational Functions
Q
:
\bf{Q:}
Q
:
Find the exact list of vertical asymptotes of
f
(
x
)
=
x
−
1
x
2
+
x
\displaystyle f\left(x\right)=\frac{x-1}{x^2+x}
f
(
x
)
=
x
2
+
x
x
−
1
Rational Functions: Slant Asymptote
Find the slant asymptote to the curve
y
=
x
3
+
x
2
+
1
x
2
+
1
\displaystyle y=\frac{x^3+x^2+1}{x^2+1}
y
=
x
2
+
1
x
3
+
x
2
+
1
.
Rational Functions: Inequalities with Fractions
Find all
x
x
x
that satisfy
x
+
3
x
2
+
8
x
+
15
≤
0
\dfrac{x+3}{x^2+8x+15}\le0
x
2
+
8
x
+
15
x
+
3
≤
0
Rational Functions
Find all
x
x
x
that satisfy
x
x
−
3
≥
4
\dfrac{x}{x-3}\ge4
x
−
3
x
≥
4
Solve the following inequality:
2
x
−
1
x
−
5
≥
x
+
1
x
+
5
\dfrac{2x-1}{x-5}\ge\dfrac{x+1}{x+5}
x
−
5
2
x
−
1
≥
x
+
5
x
+
1
Q
:
\bf{Q:}
Q
:
Find
f
(
g
(
x
)
)
f(g(x))
f
(
g
(
x
))
and its domain if
f
(
x
)
=
1
x
−
1
f\left(x\right)=\dfrac{1}{x-1}
f
(
x
)
=
x
−
1
1
and
g
(
x
)
=
1
x
+
2
g\left(x\right)=\dfrac{1}{x+2}
g
(
x
)
=
x
+
2
1
Rational Functions
Solve the following inequality:
3
x
+
1
x
−
2
≤
2
x
−
1
x
+
2
\frac{3x+1}{x-2}\leq \frac{2x-1}{x+2}
x
−
2
3
x
+
1
≤
x
+
2
2
x
−
1
Factoring and Cancelling terms in a Rational Function
When
x
=
2.1
x = 2.1
x
=
2.1
, the function
f
(
x
)
=
3
x
2
−
3
x
−
6
x
3
−
2
x
2
−
x
+
2
f(x) = \frac{3x^2 - 3x - 6}{x^3 - 2x^2 - x + 2}
f
(
x
)
=
x
3
−
2
x
2
−
x
+
2
3
x
2
−
3
x
−
6
is closest to which number?
Evaluating Rational Functions at a Point
When
x
=
−
100
x = -100
x
=
−
100
, the function
f
(
x
)
=
−
3
x
16
−
2
x
10
−
x
3
+
2
15
x
13
+
2
x
10
−
x
5
+
3
f(x) = \frac{-3 x^{16} - 2 x^{10} - x^3 + 2}{15 x^{13} + 2 x^{10} - x^5 + 3}
f
(
x
)
=
15
x
13
+
2
x
10
−
x
5
+
3
−
3
x
16
−
2
x
10
−
x
3
+
2
is closest to which number?
Rational Functions
When
x
=
0.01
x = 0.01
x
=
0.01
the function
f
(
x
)
=
3
x
9
+
7
x
3
−
10
5
x
8
−
3
x
7
+
2
x
f(x) = \frac{3x^9 + 7x^3 - 10}{5x^8 -3x^7 + 2x}
f
(
x
)
=
5
x
8
−
3
x
7
+
2
x
3
x
9
+
7
x
3
−
10
is closest to which number?
Rational Functions
When
x
=
1000
x = 1000
x
=
1000
the function
f
(
x
)
=
3
x
9
+
7
x
3
−
10
5
x
8
−
3
x
7
+
2
x
f(x) = \frac{3x^9 + 7x^3 - 10}{5x^8 -3x^7 + 2x}
f
(
x
)
=
5
x
8
−
3
x
7
+
2
x
3
x
9
+
7
x
3
−
10
is closest to which number?