Properties of Functions: Videos & Practice Problems

Understanding functions involves recognizing key properties such as maximum and minimum values, which indicate the highest and lowest points of a function, respectively. Functions can be increasing, decreasing, or constant, reflecting trends in output as input changes. Symmetry plays a crucial role, with even functions exhibiting y-axis symmetry and odd functions showing origin symmetry. The vertical line test helps determine if a relation is a function, as it ensures each input corresponds to a single output. Mastering these concepts is essential for analyzing and interpreting various functions effectively.

Properties of Functions

Understanding the properties of functions is crucial for analyzing their behavior and characteristics. One of the fundamental aspects is identifying the maximum and minimum values of a function. The maximum value refers to the highest point on the graph, indicating the greatest output (y-value) for a given input (x-value). Conversely, the minimum value represents the lowest point, where the output is less than or equal to all other outputs for the function. These points are essential for determining the overall behavior of the function.

Another important concept is the classification of functions based on their trends. A function is considered increasing if, as you move from left to right on the graph, the output values rise. In contrast, a decreasing function shows a downward trend, where the output values fall as you move along the x-axis. If the graph remains flat, it indicates a constant function, where the output remains the same regardless of the input.

Symmetry is another key property to explore. A function exhibits y-axis symmetry (even function) if folding the graph along the y-axis results in two identical halves. Mathematically, this means that for any input x, the output for -x is the same as for x, expressed as f(-x) = f(x). On the other hand, origin symmetry (odd function) occurs when folding the graph first along the y-axis and then the x-axis yields the same graph. This is represented by the equation f(-x) = -f(x), indicating that the outputs are opposites for corresponding inputs.

Lastly, x-axis symmetry can be misleading in the context of functions. While a graph may appear symmetric about the x-axis, it does not qualify as a function if a single x-value corresponds to multiple y-values. This situation fails the vertical line test, which states that if a vertical line intersects the graph at more than one point, it is not a function.

In summary, recognizing these properties—maximum and minimum values, increasing and decreasing trends, and various types of symmetry—provides a solid foundation for understanding and analyzing functions and their graphs.

Given the graph of the following function, determine the intervals on which $f\left(x\right)$  is increasing.

• A. $(-4,-3)$ and $(2,4)$
• B. $[-4,-2]$ and $[1,4]$
• C. $[-2,-1]$
• D. $(-\infty ,-2)$ and $(1,\infty )$

Answer: (D)

Given the graph of the following function, determine the intervals on which $f\left(x\right)$  is decreasing.

• A. $(-\infty ,-2)$
• B. $(-2,1)$
• C. $(-2,-\infty )$
• D. $(1,-\infty )$

Answer: (B)

Given the graph of the following function, determine where the graph reaches a maximum.

• A. $x=2$
• B. $x=-2$
• C. $x=3$
• D. $x=-4$

Answer: (A)

Here’s what students ask on this topic:

What is the maximum value of a function and how do you find it?

The maximum value of a function is the highest point on its graph. To find it, you need to identify the x-value at which the function reaches its highest output (y-value). This can be done by analyzing the graph visually or by using calculus techniques such as finding the derivative and setting it to zero to locate critical points. Then, evaluate the function at these points to determine which one gives the highest value.

What is the minimum value of a function and how do you find it?

The minimum value of a function is the lowest point on its graph. To find it, you need to identify the x-value at which the function reaches its lowest output (y-value). This can be done by analyzing the graph visually or by using calculus techniques such as finding the derivative and setting it to zero to locate critical points. Then, evaluate the function at these points to determine which one gives the lowest value.

What is the difference between increasing and decreasing functions?

An increasing function is one where the output (y-value) increases as the input (x-value) increases. Conversely, a decreasing function is one where the output decreases as the input increases. You can determine this by observing the graph: if it goes upwards from left to right, it is increasing; if it goes downwards, it is decreasing. Mathematically, a function f(x) is increasing if f'(x) > 0 and decreasing if f'(x) < 0.

What is an even function and how can you identify it?

An even function is symmetric about the y-axis. This means that for every x-value, the function's output is the same for both positive and negative x-values. Mathematically, a function f(x) is even if f(-x) = f(x) for all x in its domain. You can identify an even function by checking this condition or by visually inspecting the graph for y-axis symmetry.

What is an odd function and how can you identify it?

An odd function is symmetric about the origin. This means that for every x-value, the function's output is the negative of the output for the negative x-value. Mathematically, a function f(x) is odd if f(-x) = -f(x) for all x in its domain. You can identify an odd function by checking this condition or by visually inspecting the graph for origin symmetry.

What is the vertical line test and how is it used to determine if a relation is a function?

The vertical line test is a method used to determine if a relation is a function. To use it, draw vertical lines through the graph of the relation. If any vertical line intersects the graph at more than one point, the relation is not a function. This is because a function can only have one output (y-value) for each input (x-value).