Frequently Asked Questions about Key Features of Functions

Oct 13, 2025 by miodeya.com 100 views

Functions are a fundamental concept in mathematics, and understanding their key features is crucial for solving problems and making informed decisions. In this article, we will discuss the true statements about key features of functions, including their behavior, range, and domain.

Key Features of Functions

1.1 Decreasing Functions

A function is considered decreasing when it goes down from left to right, not from right to left. This means that as the input value increases, the output value decreases. For example, the function f(x) = -x is a decreasing function because as x increases, f(x) decreases.

Example: Consider the function f(x) = -x. As x increases from 0 to 1, f(x) decreases from 0 to -1.

1.2 Positive Functions

A function is considered positive when its graph is above the x-axis. This means that the output value is always greater than or equal to zero. For example, the function f(x) = x^2 is a positive function because its graph is always above the x-axis.

Example: Consider the function f(x) = x^2. As x increases from 0 to 1, f(x) increases from 0 to 1, and the graph remains above the x-axis.

1.3 Range of a Function

The range of a function is the set of all possible output values. It is the set of all y-values that the function can take. For example, the range of the function f(x) = x^2 is all non-negative real numbers, including zero.

Example: Consider the function f(x) = x^2. The range of this function is all non-negative real numbers, including zero.

1.4 Domain of a Function

The domain of a function is the set of all possible input values. It is the set of all x-values that the function can take. For example, the domain of the function f(x) = 1/x is all real numbers except zero.

Example: Consider the function f(x) = 1/x. The domain of this function is all real numbers except zero.

Conclusion

In conclusion, understanding the key features of functions is crucial for solving problems and making informed decisions. A function is decreasing when it goes down from left to right, not from right to left. A function is positive when its graph is above the x-axis. The range of a function is the set of all possible output values, and the domain of a function is the set of all possible input values.

Key Takeaways:

A function is decreasing when it goes down from left to right.
A function is positive when its graph is above the x-axis.
The range of a function is the set of all possible output values.
The domain of a function is the set of all possible input values.

Recommended Reading:

Algebra and Functions by Michael Artin
Calculus by Michael Spivak
Functions and Graphs by James Callahan

Practice Problems:

Determine whether the function f(x) = -x is increasing or decreasing.
Determine whether the function f(x) = x^2 is positive or negative.
Find the range and domain of the function f(x) = 1/x.

Solutions:

The function f(x) = -x is decreasing.
The function f(x) = x^2 is positive.
The range of the function f(x) = 1/x is all non-zero real numbers.
The domain of the function f(x) = 1/x is all real numbers except zero.
Frequently Asked Questions about Key Features of Functions ===========================================================

Functions are a fundamental concept in mathematics, and understanding their key features is crucial for solving problems and making informed decisions. In this article, we will answer some frequently asked questions about key features of functions.

Q: What is a decreasing function?

A: A function is considered decreasing when it goes down from left to right, not from right to left. This means that as the input value increases, the output value decreases.

Q: What is a positive function?

A: A function is considered positive when its graph is above the x-axis. This means that the output value is always greater than or equal to zero.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values. It is the set of all y-values that the function can take.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values. It is the set of all x-values that the function can take.

Q: How do I determine whether a function is increasing or decreasing?

A: To determine whether a function is increasing or decreasing, you can use the following steps:

Choose a point on the graph of the function.
Move to the right of the point and check if the function value increases or decreases.
If the function value increases, the function is increasing. If the function value decreases, the function is decreasing.

Q: How do I determine whether a function is positive or negative?

A: To determine whether a function is positive or negative, you can use the following steps:

Choose a point on the graph of the function.
Check if the function value is above or below the x-axis.
If the function value is above the x-axis, the function is positive. If the function value is below the x-axis, the function is negative.

Q: How do I find the range and domain of a function?

A: To find the range and domain of a function, you can use the following steps:

Determine the type of function (linear, quadratic, polynomial, etc.).
Use the properties of the function to determine the range and domain.
Check the graph of the function to see if it is above or below the x-axis.

Q: What are some common types of functions?

A: Some common types of functions include:

Linear functions: f(x) = mx + b
Quadratic functions: f(x) = ax^2 + bx + c
Polynomial functions: f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0
Rational functions: f(x) = p(x) / q(x)

Q: How do I graph a function?

A: To graph a function, you can use the following steps:

Choose a point on the graph of the function.
Move to the right of the point and check if the function value increases or decreases.
Use a ruler or a graphing tool to draw the graph of the function.