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Rational Functions

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Let f(x)=x2+5x+6x−4f(x)=\dfrac{x^2+5x+6}{x-4} be a rational function.


Identify the following information:

More Rational Functions Questions:
Rational Functions: Slant Asymptote
Find the slant asymptote to the curve y=x3+x2+1x2+1\displaystyle y=\frac{x^3+x^2+1}{x^2+1}.
Rational Functions
Q:\bf{Q:} Find the exact list of vertical asymptotes of f(x)=x−1x2+x\displaystyle f\left(x\right)=\frac{x-1}{x^2+x}
Rational Functions: Slant Asymptote
Find the slant asymptote to the curve y=x3+x2+1x2+1\displaystyle y=\frac{x^3+x^2+1}{x^2+1}.
Q:\bf{Q:} Find f(g(x))f(g(x)) and its domain if f(x)=1x−1f\left(x\right)=\dfrac{1}{x-1} and g(x)=1x+2g\left(x\right)=\dfrac{1}{x+2}
Solve the following inequality: 2x−1x−5≥x+1x+5\dfrac{2x-1}{x-5}\ge\dfrac{x+1}{x+5}
Rational Functions: Inequalities with Fractions
Find all xx that satisfy x+3x2+8x+15≤0\dfrac{x+3}{x^2+8x+15}\le0
Rational Functions
Find all xx that satisfy xx−3≥4\dfrac{x}{x-3}\ge4
Rules of Exponents
Simplify: −5x2ya33z5⋅37xby15z−2-\dfrac{5x^2y^a}{3^3z^5}\cdot\dfrac{3^7x^by}{15z^{-2}}
Solving Rational Equations
Practice: Solving Rational Equations
Let f(x)=4x2−252x+5f(x)=\dfrac{4x^2-25}{2x+5}.
If f(x)=4x2−2x+1f(x)=4x^2-2x+1, then what is x?x?
Solving Rational Equations
Practice: Solving Rational Equations
Let f(x)=x2−4x−2f(x)=\dfrac{x^2-4}{x-2}.
If f(x)=x2−5x−6f(x)=x^2-5x-6, then what is x?x?
Solving Rational Equations
Practice: Solving Rational Equations
Given the equation 4xx−5+3x+2=xx2−3x−10\dfrac{4x}{x-5}+\dfrac{3}{x+2}=\dfrac{x}{x^2-3x-10}, answer the following questions.
Solving Rational Equations
Practice: Solving Rational Equations
Given the equation 4x+1−1x+3=x+2x2+4x+3\dfrac{4}{x+1}-\dfrac{1}{x+3}=\dfrac{x+2}{x^2+4x+3}, answer the following questions.
Solving Rational Equations
Practice: Solving Rational Equations
True or false:
8126\dfrac{81}{26} is a solution to 10=4x−13x+810=\dfrac{4x-1}{3x+8}.
Solving Rational Equations
Practice: Solving Rational Equations
True or false:
177\dfrac{17}{7} is a solution to −5=2x+3x−4-5=\dfrac{2x+3}{x-4}.
Solving Rational Equations
Solve for xx:
52x−1−73x+2=96x2+x−2\dfrac{5}{2x-1}-\dfrac{7}{3x+2}=\dfrac{9}{6x^2+x-2}
Solving Rational Equations
Solve 2x−3x2−3=5x+1\dfrac{2x-3}{x^2-3}=\dfrac{5}{x+1}.
Graphing Rational Functions
Practice: Graphing Rational Functions
Write a rational function with the following properties:
An oblique asymptote at y=−3x+1y=-3x+1
Graphing Rational Functions
Practice: Graphing Rational Functions
Write a rational function with the following properties:
An oblique asymptote at y=2x−7y=2x-7
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−1x=-1
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−2x=-2
Graphing Rational Functions
Practice: Graphing Rational Functions
Sketch a graph of f(x)=x2−25x+5f(x)=\dfrac{x^2-25}{x+5}, 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+9x+3f(x)=\dfrac{-x+9}{x+3}, 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=0y=0.
Graphing Rational Functions
Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at y=0y=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)=3x2+14x+8x2−2x−14f(x)=\dfrac{3x^2+14x+8}{x^2-2x-14}
Rational Functions
Practice: Rational Functions
Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of:
f(x)=2x2+x−12x2+5x−3f(x)=\dfrac{2x^2+x-1}{2x^2+5x-3}
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)=x2−3xx+6f(x) = \frac{x^2 - 3x}{x + 6} for xx on the small interval [−1/10,1/10][-1/10, 1/10]
Rational Functions
Match the following five functions with the above graphs.
Rules of Exponents
Simplify: −5x2ya33z5⋅37xby15z−2-\dfrac{5x^2y^a}{3^3z^5}\cdot\dfrac{3^7x^by}{15z^{-2}}
Rational Functions
Q:\bf{Q:} Find the exact list of vertical asymptotes of f(x)=x−1x2+x\displaystyle f\left(x\right)=\frac{x-1}{x^2+x}
Rational Functions: Slant Asymptote
Find the slant asymptote to the curve y=x3+x2+1x2+1\displaystyle y=\frac{x^3+x^2+1}{x^2+1}.
Rational Functions: Inequalities with Fractions
Find all xx that satisfy x+3x2+8x+15≤0\dfrac{x+3}{x^2+8x+15}\le0
Rational Functions
Find all xx that satisfy xx−3≥4\dfrac{x}{x-3}\ge4
Solve the following inequality: 2x−1x−5≥x+1x+5\dfrac{2x-1}{x-5}\ge\dfrac{x+1}{x+5}
Q:\bf{Q:} Find f(g(x))f(g(x)) and its domain if f(x)=1x−1f\left(x\right)=\dfrac{1}{x-1} and g(x)=1x+2g\left(x\right)=\dfrac{1}{x+2}
Rational Functions
Solve the following inequality: 3x+1x−2≤2x−1x+2\frac{3x+1}{x-2}\leq \frac{2x-1}{x+2}
Rational Functions
Solve the following inequality for xx
x−1x+1≤x+2x−1\frac{x-1}{x+1} \leq \frac{x+2}{x-1}
Factoring and Cancelling terms in a Rational Function
When x=2.1x = 2.1 , the function
f(x)=3x2−3x−6x3−2x2−x+2f(x) = \frac{3x^2 - 3x - 6}{x^3 - 2x^2 - x + 2}
is closest to which number?
Evaluating Rational Functions at a Point
When x=−100x = -100 , the function
f(x)=−3x16−2x10−x3+215x13+2x10−x5+3f(x) = \frac{-3 x^{16} - 2 x^{10} - x^3 + 2}{15 x^{13} + 2 x^{10} - x^5 + 3}
is closest to which number?
Rational Functions
When x=0.01x = 0.01 the function
f(x)=3x9+7x3−105x8−3x7+2xf(x) = \frac{3x^9 + 7x^3 - 10}{5x^8 -3x^7 + 2x}
is closest to which number?
Rational Functions
When x=1000x = 1000 the function
f(x)=3x9+7x3−105x8−3x7+2xf(x) = \frac{3x^9 + 7x^3 - 10}{5x^8 -3x^7 + 2x}
is closest to which number?
More Graphs of Rational Functions Questions:
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−1x=-1
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−2x=-2
Graphing Rational Functions
Practice: Graphing Rational Functions
Sketch a graph of f(x)=x2−25x+5f(x)=\dfrac{x^2-25}{x+5}, 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+9x+3f(x)=\dfrac{-x+9}{x+3}, finding all asymptotes, intercepts, missing points, and positive & negative intervals.
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.
More Rational Functions Questions:
Graphing Rational Functions
Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at y=0y=0.
Graphing Rational Functions
Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at y=0y=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)=3x2+14x+8x2−2x−14f(x)=\dfrac{3x^2+14x+8}{x^2-2x-14}
Rational Functions
Practice: Rational Functions
Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of:
f(x)=2x2+x−12x2+5x−3f(x)=\dfrac{2x^2+x-1}{2x^2+5x-3}
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.
More Graphing Rational Functions Questions:
Graphing Rational Functions
Practice: Graphing Rational Functions
Write a rational function with the following properties:
An oblique asymptote at y=−3x+1y=-3x+1
Graphing Rational Functions
Practice: Graphing Rational Functions
Write a rational function with the following properties:
An oblique asymptote at y=2x−7y=2x-7
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−1x=-1
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−2x=-2
Graphing Rational Functions
Practice: Graphing Rational Functions
Sketch a graph of f(x)=x2−25x+5f(x)=\dfrac{x^2-25}{x+5}, 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+9x+3f(x)=\dfrac{-x+9}{x+3}, finding all asymptotes, intercepts, missing points, and positive & negative intervals.
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.
More Graphs of Rational Functions Questions:
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−1x=-1
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−2x=-2
Graphing Rational Functions
Practice: Graphing Rational Functions
Sketch a graph of f(x)=x2−25x+5f(x)=\dfrac{x^2-25}{x+5}, 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+9x+3f(x)=\dfrac{-x+9}{x+3}, finding all asymptotes, intercepts, missing points, and positive & negative intervals.
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.
More Graphs of Rational Functions Questions:
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−1x=-1
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−2x=-2
Graphing Rational Functions
Practice: Graphing Rational Functions
Sketch a graph of f(x)=x2−25x+5f(x)=\dfrac{x^2-25}{x+5}, 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+9x+3f(x)=\dfrac{-x+9}{x+3}, finding all asymptotes, intercepts, missing points, and positive & negative intervals.
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.
Graphing Reciprocal Functions
Practice: Graphing Reciprocal Functions
Determine the equation for the function graphed below:
Graphing Reciprocal Functions
Practice: Graphing Reciprocal Functions
Determine the equation for the function graphed below:
Graphing Reciprocal Functions
Practice: Graphing Reciprocal Functions
Sketch a graph of y=−2x2+4y=-2x^2+4 and it's reciprocal on the same grid, labeling vertical asymptotes.
Graphing Reciprocal Functions
Practice: Graphing Reciprocal Functions
Sketch a graph of the reciprocal ofy=(x+2)2−2y=(x+2)^2-2 .
Graphing Reciprocal Functions
Practice: Graphing Reciprocal Functions
Sketch a graph of the reciprocal of the function below:
Graphing Reciprocal Functions
Practice: Graphing Reciprocal Functions
Sketch a graph of the reciprocal of the function below:
More Rational Functions Questions:
Graphing Rational Functions
Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at y=0y=0.
Graphing Rational Functions
Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at y=0y=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)=3x2+14x+8x2−2x−14f(x)=\dfrac{3x^2+14x+8}{x^2-2x-14}
Rational Functions
Practice: Rational Functions
Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of:
f(x)=2x2+x−12x2+5x−3f(x)=\dfrac{2x^2+x-1}{2x^2+5x-3}
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.
More Graphing Rational Functions Questions:
Graphing Rational Functions
Practice: Graphing Rational Functions
Write a rational function with the following properties:
An oblique asymptote at y=−3x+1y=-3x+1
Graphing Rational Functions
Practice: Graphing Rational Functions
Write a rational function with the following properties:
An oblique asymptote at y=2x−7y=2x-7
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−1x=-1
Graphing Rational Functions
Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d} that describes the following:
Vertical asymptote at x=−2x=-2
Graphing Rational Functions
Practice: Graphing Rational Functions
Sketch a graph of f(x)=x2−25x+5f(x)=\dfrac{x^2-25}{x+5}, 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+9x+3f(x)=\dfrac{-x+9}{x+3}, finding all asymptotes, intercepts, missing points, and positive & negative intervals.
Rational Functions
Practice: Rational Functions
Match the rational function with the correct sketch of its graph.
More Rational Functions Questions:
Graphing Rational Functions
Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at y=0y=0.
Graphing Rational Functions
Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at y=0y=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)=3x2+14x+8x2−2x−14f(x)=\dfrac{3x^2+14x+8}{x^2-2x-14}
Rational Functions
Practice: Rational Functions
Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of:
f(x)=2x2+x−12x2+5x−3f(x)=\dfrac{2x^2+x-1}{2x^2+5x-3}
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.