Master function concepts with step-by-step practice problems. Learn to evaluate functions, identify domains, and understand function notation through interactive exercises.
On one hand, functions are a fairly abstract concept, but on the other hand, they are very useful in many areas of mathematics. The topic of functions dominates many fields, including algebra, trigonometry, differential and integral calculus, and more. Therefore, it's important to understand the concept of functions, so that it can be applied in any of the fields of mathematics, and especially when we start learning about functions in seventh grade.
A function expresses a relationship between two variables (X and Y)
An independent variable is a non-variable constant by which we explain , the dependent variable
For example, if Daniela worked as a babysitter and earned 30 dollars per hour and we want to know how much Daniela made after hours, the number of hours worked is actually the independent variable with which we know how much she earned. Ultimately this is the dependent variable.
In other words, it can be said that the amount Daniela earned is a function of the number of hours she worked .
We will mark the data of the function algebraically in this way:
It's important to remember that each element in the domain will always have only one element in the range .
This means that it's not possible that during the hours Daniela worked, she received both dollars and dollars.
In which interval does the function decrease?
Red line: \( x=0.65 \)
In what domain does the function increase?
Let's remember that the function increases if the values and values increase simultaneously.
On the other hand, the function decreases if the values increase while the values decrease simultaneously.
In the given graph, we can see that the function increases in the domain where x > 0 ; in other words, where the values are increasing.
Answer:
x > 0
Determine in which domain the function is negative?
Remember that a function is increasing if both X values and Y values are increasing simultaneously.
A function is decreasing if X values are increasing while Y values are decreasing simultaneously.
In the graph, we can observe that in the domain x > 1 the function is decreasing, meaning the Y values are decreasing.
Answer:
x > 1
In what domain is the function increasing?
Let's first remember that a function is increasing if both the X and Y values are increasing simultaneously.
Conversely, a function is decreasing if the X values are increasing while the Y values are decreasing simultaneously.
In the graph shown, we can see that the function is increasing in every domain and therefore the function is increasing for all values of X.
Answer:
All values of
In what domain does the function increase?
Let's remember that the function increases if the X values and Y values increase simultaneously.
On the other hand, the function decreases if the X values increase and the Y values decrease simultaneously.
In the given graph, we notice that the function increases in the domain where x < 0 , meaning the Y values are increasing.
Answer:
x<0
In what interval is the function increasing?
Purple line:
Let's remember that a function is described as increasing if both X values and Y values are increasing simultaneously.
A function is decreasing if X values are increasing while Y values are decreasing simultaneously.
In the graph, we can see that in the domain x < 0.6 the function is increasing, meaning the Y values are increasing.
Answer:
x<0.6