Find Domain & Sketch Graph: Calculus Functions

In summary, the function p(t) = |2t + 3| must be split into a piecewise function due to the absolute value not allowing for negative values. The first part of the function is greater than or equal to zero, while the second part is less than zero. This is because the absolute value keeps the value positive, so when the input is negative, an additional negative sign is needed to make it positive. The function can also be understood as a transformation of the function f(x) = |x|, where the factor of two makes it twice as narrow and the translation of +3/2 moves it to the left by 3/2 units.
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Homework Statement


Find the Domain and sketch the graph of the function if a grid is provided.

1. p(t) = | 2t + 3 |


Homework Equations





The Attempt at a Solution



Well basically in the book it said that it would end up looking something like this... {2t + 3 for (2t + 3) > 0 (<----- that is suppost to be greater or equal to)

and the other part was suppost to be { -(2t + 3) for (2t + 3) < 0


to me this makes no sense of why this is suppost to end up this way. Also ignore the part that said if you're provided a grid. You don't have to solve it or anything like that, but just explain this to me. Thank you, Corkery.
 
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  • #2
Since the absolute value does not allow for negatives, the function must be split into the piecewise function of the part where it is negative, and the part where it is positive.
 
  • #3
Mindscrape said:
Since the absolute value does not allow for negatives, the function must be split into the piecewise function of the part where it is negative, and the part where it is positive.

why is one of the equations greater than or equal to zero and another is less then zero. I just don't understand why you would do this. Is there a lamens term explanation, haha or is it already in lamens terms?
 
  • #4
Basically, let 2t+3 = u

Absolute value signs mean, keep the u positive at all costs! RAWR!

so, when u is negative, less than zero, to make it postive stick another negative sign in front right?!

When u is positive, leave it as it is.

So that splits it into 2 bits thatll keep it positive!

-(2t+3) for < 0, 2t+3 for > 0 :D:D
 
  • #5
WOW! I actually understand that! thank you so much GIB Z and thank you Mindscrape. You guys rock!
 
  • #6
or you can "read" the function

let f(x)=|x|

basically, you are asked to figure out the graph for
f(2(t+3/2))

the factor of two means 2 times narrower, +3/2 means translation to the left for 3/2 units.

just transform the V shape according to the recipe and you are done.
 

1. What is the purpose of finding the domain and sketching a graph of a calculus function?

Finding the domain and sketching the graph of a calculus function is important because it helps us understand the behavior of the function and identify any potential issues or restrictions. It also allows us to visualize the function and make predictions about its behavior.

2. How do you find the domain of a calculus function?

To find the domain of a calculus function, we need to identify all possible values of the independent variable (usually x) that make the function defined and meaningful. This means looking for any potential restrictions such as division by zero, negative values under a square root, or negative values in the denominator of a fraction.

3. What is a vertical asymptote and how does it affect the domain of a calculus function?

A vertical asymptote is a vertical line on a graph that a function approaches but never touches. It is usually caused by a value of the independent variable that makes the function undefined. A vertical asymptote affects the domain of a calculus function by limiting the values of the independent variable that the function can take. Any values that would result in a vertical asymptote are not included in the domain.

4. How do you sketch a graph of a calculus function?

To sketch a graph of a calculus function, first find the domain and any intercepts or asymptotes. Then, choose a few values of the independent variable and calculate the corresponding values of the function. Plot these points on a coordinate plane and connect them with a smooth curve. You can also use a graphing calculator to help you visualize the graph.

5. Can the domain of a calculus function change?

Yes, the domain of a calculus function can change depending on the restrictions of the function. For example, if the function contains a square root, the domain would exclude any negative values under the square root. If another function is added or subtracted to the original function, the domain may also change accordingly. It is important to always check for any potential changes in the domain when dealing with calculus functions.

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