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General Mathematics Activity (09-06-24)

11 min read

FunctionDomainRange-intercept-interceptVertical AsymptoteHorizontal Asymptote

No H.A
No V.A.No H.A

Domain

To find the domain, locate the denominator , then equate to .

Remove in the domain, therefore:

Range

To find the range, swap the and variables.

Then, solve for to get the inverse function.

After this, we solve for the domain.

Solving the domain of the inverse allows us to get the range of the original function.

Locate the denominator of the inverse, , then equate to .

We exclude this to the range, therefore:

-intercept

To find the -intercept, locate the numerator, , then equate to .

Therefore, is the -intercept.

-intercept

Evaluate when .

Therefore, is the -intercept.

Vertical Asymptote

The excluded values in the domain are the same thing as the positions of the vertical asymptotes.

Therefore, is a vertical asymptote.

Horizontal Asymptote

Since the degree of the numerator is and the degree of the denominator is , then they have the same degree. To get the horizontal asymptote:

The leading coefficient of the numerator is .

The leading coefficient of the denominator is ,


Therefore, the horizontal asymptote is , or simply .

Domain

To find the domain, locate the denominator , then equate to .

Remove in the domain, therefore:

Range

To find the range, swap the and variables.

Then, solve for to get the inverse function.

After this, we solve for the domain.

Solving the domain of the inverse allows us to get the range of the original function.


Locate the denominator of the inverse, , then equate to .

We exclude this to the range, therefore:

-intercept

To find the -intercept, locate the numerator, , then equate to .

Therefore, is the -intercept.

-intercept

Evaluate when .

Therefore, is the -intercept.

Vertical Asymptote

The excluded values in the domain are the same thing as the positions of the vertical asymptotes.

Therefore, is a vertical asymptote.

Horizontal Asymptote

Since the degree of the numerator is and the degree of the denominator is , then they have the same degree. To get the horizontal asymptote:

The leading coefficient of the numerator is .

The leading coefficient of the denominator is ,


Therefore, the horizontal asymptote is ,

Domain

To get the domain, simply set the denominator equal to .

Remove in the domain, therefore:

Range

To find the range, swap the and variables.

Then, solve for to get the inverse function.

Since the variable has a second degree term, we rewrite the equation as .

To isolate the term, we apply the quadratic formula in terms of .

We then solve for the domain of this function.

Since this function is the inverse of the original function, then finding the domain of this function should give us the range of the original function.

This function will only be defined when the expression under the radical sign is greater than or equal to zero.

We apply rules of quadratic inequalities.

The only way for an upward facing parabola be greater than is if is less than the smaller root and is greater than the larger root .

Therefore, the range falls under the interval:

-intercept

Take the numerator , then equate to .

Then we use zero product property:

If one factor is zero, then everything else is zero

  • For , then .
  • For , then .

Therefore, the -intercepts are and .

-intercept

For the -intercept, simply evaluate .

Therefore, the -intercept is

Vertical Asymptote

The restrictions in the domain are the same thing as the positions of the asymptotes.

Since is not included in the domain, must be a vertical asymptote.

Horizontal Asymptote

Since the degree of the numerator is greater than the degree of the denominator , then this function does not have a horizontal asymptote.

Domain

To find the domain, take the denominator , and set it equal to .

must be excluded in the domain.

Therefore, the domain is:

Range

To find the range, we can simply consider analyzing the function itself

Factor into .

Then, we can cancel from the numerator and the denominator.

The domain and range of is all real numbers.

However, remember that we have a domain restriction at .

Since is one-to-one, we can evaluate to get the range restriction for the original function.

Therefore, the range is:

-intercept

To find the -intercept, we can simply equate the simplified linear expression that we got to 0.

Therefore, the -intercept is .

-intercept

To find the -intercept, simply evaluate .

Therefore, the -intercept is

Vertical Asymptote

To find the vertical asymptotes, we look when the denominator is .

Since cancels with the in the numerator, then there is no vertical asymptote and instead, we have a removable discontinuity.

Horizontal Asymptote

Since the degree of the numerator is greater than the degree of the denominator , then this function does not have a horizontal asymptote.

Domain

Set the denominator equal to .

Exclude them in the domain.

Using zero product property:

  • gives us , and
  • gives us .

Therefore, the domain is:

Range

To find the range, swap the variables.

Then, solve for to get the inverse function.

Since the variable has a second degree term, we rewrite the equation as .

To isolate the term, we apply the quadratic formula in terms of .

We then solve for the domain of this function.

Since this function is the inverse of the original function, then finding the domain of this function should give us the range of the original function.


This function will only be defined when the expression under the radical sign is greater than or equal to zero.

We apply rules of quadratic inequalities.

The only way for an upward facing parabola be greater than is if is less than the smaller root and is greater than the larger root .

Therefore, the range falls under the interval:

-intercept

Take the numerator, , then set equal to .

Therefore, the -intercept is

-intercept

To find the -intercept, simply evaluate

Therefore, -intercept is

Vertical Asymptote

The vertical asymptotes are the same thing as the restrictions in the domain, given that no factors cancel in the numerator and the denominator.

Since, and are included in the domain, therefore, the vertical asymptotes are and .

Horizontal Asymptote

Since the degree of the numerator is 1 and the degree of the denominator is , then the degree of the numerator is less than the degree of the denominator.

Therefore, the horizontal asymptote is .

Graph View