+0

# Rational Functions for Algebra 2

+2
332
5
+-13

I missed a few lessons and I need help:

Can someone give me an example with steps to help me understand how to find Factored/Simplified Equation, Removable Discontinuity, Vertical and Horizontal Asymptote, domain/range, and graph for an equation in the format of g(x)=something/something

I have a few examples from my homework, but if we're not allowed to have y'all do it for me, just give me an example so I won't be plagarizing.

Here they are:

g(x)=1/x^2-4

h(x)=x+3/x^2+5x+4

h(x)=x^2-9/(x+3)

f(x)=2x+1/x

y=x^2+8x+15/x+5

Apr 20, 2019
edited by Guest  Apr 20, 2019
edited by TheDerivativ  Apr 20, 2019

#4
+8341
+3

Factored/Simplified Expressions

You take an expression, find all its factors, write it out as a product of factors, you get the factored form.

You expand all the product, then you get the simplified form.

Example:

$$x^2+2x-15\qquad \text{This is}\color{blue}{\text{ simplified}}\color{black}\text{.}\\ (x+5)(x-3)\qquad \text{This is}\color{magenta}{\text{ factored}}\color{black}\text{.}\\$$

Removable discontinuity

This usually only happens when the given function is rational.

1. Find the G.C.D. of the numerator and the denominator.

2. Set the G.C.D. to 0

3. The roots of the equation in step 2 is where the removable discontinuity occurs.

Example:

Find the removable discontinuity(discontinuities) of the function $$f(x) = \dfrac{\dfrac{1}{4}x^2-x+1}{x^2-4}$$.

Solution:

$$f(x)\\ =\dfrac{\dfrac{1}{4}x^2-x+1}{x^2-4}\\ =\dfrac{\dfrac{1}{4}(x^2-4x+4)}{(x-2)(x+2)}\\ =\dfrac{\dfrac{1}{4}(x-2)^2}{(x-2)(x+2)}\\ =\dfrac{(x-2)^2}{4(x-2)(x+2)}$$

GCD of numerator and denominator is (x-2). <- Set this to 0.

x - 2 = 0

x = 2

The removable discontinuity of this function occurs at x = 2.

Vertical asymptotes and horizontal asymptotes

Vertical asymptotes occurs at (unremovable) discontinuities.

Horizontal asymptotes is just a horizontal line which the function approaches when x reaches positive or negative infinity.

How to find...

Vertical asymptotes? Find all the discontinuities, exclude the removable ones and the "broken" points(like this one below, at x = 1.)

Horizontal asymptotes? That's a calculus thing. If you want to do it the algebra way, you can substitute larger and larger numbers into the function and see if the function reaches a value eventually, and then repeat with negative numbers.

Example:

Find all the horizontal and vertical asymptotes of the function y = 1/x.

Solution:

Let's find the horizontal asymptotes first. If you substitute large numbers, you will find that it approaches 0 eventually. Same for negative numbers. So the only horizontal asymptote, in this case, is y = 0.

The function has a discontinuity at x = 0, and it is not removable. So vertical asymptote is at x = 0.

Domain and range

Domain is the set of possible inputs of the function.

Range is the set of possible outputs of the function.

Example:

Find the domain and range of the function $$f(x) = \dfrac{1}{\sqrt{x-5}}$$.

Solution:

This requires some kind of sense of mathematics. Firstly, the denominator can't be 0, or it will give infinity. So we have $$\sqrt{x-5} \neq 0 \implies x - 5 \neq 0$$.

Also, $$\sqrt{ }$$ can't take negative numbers as input. So $$x - 5 \geq 0$$. But actually as the denominator can't be 0, x - 5 can't be 0. So $$x - 5 > 0\implies x > 5$$.

So the domain of the function $$\text{D}\{ f(x)\}$$ is $$(5,\infty)$$.

Now let's consider the output of the function. $$\sqrt{ }$$ only gives out nonnegative numbers(0 and positive numbers if you don't know what are nonnegative numbers), So $$\sqrt{x-5} \ge 0 \implies f(x) = \dfrac{1}{\sqrt{x-5}} \le \infty$$, which doesn't make any sense. But $$f(x) = \dfrac{1}{\text{a nonnegative number}}$$, considering that, we get the range of f(x) as the set of all positive real numbers.

$$\text{R}\{f(x)\}$$ is $$(0,\infty)$$.

P.S.: How do I know these without knowing all the terminologies? Google it ( ͡° ͜ʖ ͡°).

Apr 20, 2019
edited by MaxWong  Apr 20, 2019

#1
-13
0

Help

Apr 20, 2019
edited by TheDerivativ  Apr 20, 2019
#2
-13
0

By the way, I'm not looking for yall to answer these homework questions for me I just want a step by step method on how I would be able to work them out myself. Khan Academy is useless to me and can go die in a hole. Someone pls help and explain :/

Apr 20, 2019
#3
-2

If Khan Academy is useless to you then you will never learn anything on here.

Quit fucking around!

Apr 20, 2019
#4
+8341
+3

Factored/Simplified Expressions

You take an expression, find all its factors, write it out as a product of factors, you get the factored form.

You expand all the product, then you get the simplified form.

Example:

$$x^2+2x-15\qquad \text{This is}\color{blue}{\text{ simplified}}\color{black}\text{.}\\ (x+5)(x-3)\qquad \text{This is}\color{magenta}{\text{ factored}}\color{black}\text{.}\\$$

Removable discontinuity

This usually only happens when the given function is rational.

1. Find the G.C.D. of the numerator and the denominator.

2. Set the G.C.D. to 0

3. The roots of the equation in step 2 is where the removable discontinuity occurs.

Example:

Find the removable discontinuity(discontinuities) of the function $$f(x) = \dfrac{\dfrac{1}{4}x^2-x+1}{x^2-4}$$.

Solution:

$$f(x)\\ =\dfrac{\dfrac{1}{4}x^2-x+1}{x^2-4}\\ =\dfrac{\dfrac{1}{4}(x^2-4x+4)}{(x-2)(x+2)}\\ =\dfrac{\dfrac{1}{4}(x-2)^2}{(x-2)(x+2)}\\ =\dfrac{(x-2)^2}{4(x-2)(x+2)}$$

GCD of numerator and denominator is (x-2). <- Set this to 0.

x - 2 = 0

x = 2

The removable discontinuity of this function occurs at x = 2.

Vertical asymptotes and horizontal asymptotes

Vertical asymptotes occurs at (unremovable) discontinuities.

Horizontal asymptotes is just a horizontal line which the function approaches when x reaches positive or negative infinity.

How to find...

Vertical asymptotes? Find all the discontinuities, exclude the removable ones and the "broken" points(like this one below, at x = 1.)

Horizontal asymptotes? That's a calculus thing. If you want to do it the algebra way, you can substitute larger and larger numbers into the function and see if the function reaches a value eventually, and then repeat with negative numbers.

Example:

Find all the horizontal and vertical asymptotes of the function y = 1/x.

Solution:

Let's find the horizontal asymptotes first. If you substitute large numbers, you will find that it approaches 0 eventually. Same for negative numbers. So the only horizontal asymptote, in this case, is y = 0.

The function has a discontinuity at x = 0, and it is not removable. So vertical asymptote is at x = 0.

Domain and range

Domain is the set of possible inputs of the function.

Range is the set of possible outputs of the function.

Example:

Find the domain and range of the function $$f(x) = \dfrac{1}{\sqrt{x-5}}$$.

Solution:

This requires some kind of sense of mathematics. Firstly, the denominator can't be 0, or it will give infinity. So we have $$\sqrt{x-5} \neq 0 \implies x - 5 \neq 0$$.

Also, $$\sqrt{ }$$ can't take negative numbers as input. So $$x - 5 \geq 0$$. But actually as the denominator can't be 0, x - 5 can't be 0. So $$x - 5 > 0\implies x > 5$$.

So the domain of the function $$\text{D}\{ f(x)\}$$ is $$(5,\infty)$$.

Now let's consider the output of the function. $$\sqrt{ }$$ only gives out nonnegative numbers(0 and positive numbers if you don't know what are nonnegative numbers), So $$\sqrt{x-5} \ge 0 \implies f(x) = \dfrac{1}{\sqrt{x-5}} \le \infty$$, which doesn't make any sense. But $$f(x) = \dfrac{1}{\text{a nonnegative number}}$$, considering that, we get the range of f(x) as the set of all positive real numbers.

$$\text{R}\{f(x)\}$$ is $$(0,\infty)$$.

P.S.: How do I know these without knowing all the terminologies? Google it ( ͡° ͜ʖ ͡°).

MaxWong Apr 20, 2019
edited by MaxWong  Apr 20, 2019