Determine the domain and range of each function. Use various limits to find the asymptotes and t

Determine the domain and range of each function. Use various...

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Limit Analysis

This concept is about evaluating how a function behaves as the input approaches a specific value (such as where the function is undefined) or as it approaches infinity. It is an essential tool for identifying asymptotic behavior and understanding the boundaries of the function's range.

Domain of a Function

This concept involves determining all input values for which a function is defined. In rational functions, it requires identifying values that make the denominator zero and excluding them from the domain, ensuring that division by zero does not occur.

Range of a Function

This concept pertains to finding all possible output values of a function. For rational functions, one often needs to solve or rearrange the function equation and consider its behavior as the independent variable varies over the domain, thereby identifying any values that the function cannot assume.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. For rational functions, vertical asymptotes occur where the denominator is zero (and the function is undefined), while horizontal asymptotes are determined by analyzing the limits as the independent variable approaches positive or negative infinity.

Transcript

00:01 Okay, so we're looking for the domain and range of this function, x cubed, x cubed, over x cubed minus 8.

00:15 If we factor this, this is x cubed over x minus 2 times x squared plus 2x plus 4.

00:26 I'm just using the difference of, uh, difference of cubes equation.

00:32 And so actually we know that this polynomial is never zero.

00:43 So what do we have here? well, we have clearly a vertical asymptote at x equals 2.

00:54 And let's take the limits.

00:55 So as x approaches, say 2 from the right, y is going to be approaching, this is going to be positive, this is going to be positive, and this is going to be a small positive number, so this is approaching infinity.

01:18 And then as x approaches two from the left, this will be positive still.

01:25 This will still be positive, clearly, and this will be a small negative number.

01:29 So y will be going to minus infinity.

01:33 So we're hitting potentially every single y value in the range, except for possibly at a horizontal asymptote.

01:43 So we know at 2, we are shooting off to infinity, shooting off to negative infinity.

01:51 So because we have a vertical asymptote there, we're probably hitting every value, except for maybe some potential horizontal asymptote.

02:02 So let's think about the horizontal asymptote...

Please give Ace some feedback