Nonlinear functions are functions in math where the relationship between the input (like x) and output (like y) is not constant. This means the change in output does not happen at the same rate as the input. When you draw these functions on a graph, they do not form a straight line—they usually make a curve or other shapes.

Unlike linear functions, which grow or shrink at the same rate and always make straight lines on graphs, nonlinear functions can grow or shrink in different ways. These include polynomial functions, trigonometric functions like sine and cosine, exponential functions, and logarithmic functions.

We use nonlinear functions in many real-world areas like science, engineering, economics, and biology to show complex relationships. In computer science and machine learning, they help in understanding and predicting patterns that can’t be explained by simple straight lines.

In this article, we will learn what nonlinear functions are, how their equations, tables, and graphs look, and also go through some solved examples to understand them better.

What is a Nonlinear Function?

In math, a function is a rule that links each input to one output. The set of all possible inputs is called the domain, and the set of all possible outputs is called the range.

A nonlinear function is a function where the output does not change at a constant rate when the input changes. This means it does not make a straight line on a graph. Instead, the graph of a nonlinear function can be curved or take different shapes.

Unlike linear functions (which form straight lines), nonlinear functions can look like curves, waves, or exponential growth lines. One example of a nonlinear function is a quadratic function, which makes a U-shaped curve called a parabola.

Let’s look at a real-life example: Imagine there are 100 fishes in a pond, and the number of fishes doubles every week. This situation can be shown with the function:
f(x) = 100 × (2)^x,
where x is the number of weeks, and f(x) gives the total number of fishes.

This is an exponential function, and it's a type of nonlinear function because the number of fishes grows faster and faster each week.

To visualize this function, we can create a table and plot its graph accordingly.

Let's graph the table now.

The graph shown above does not display a straight line, indicating that it represents a nonlinear function. Nonlinear functions are characterized by non-uniform slopes. They can be defined through a table of values, an equation, or a graph. For instance, examples of nonlinear functions include polynomial functions, quadratic functions, and cubic functions.

Nonlinear Function Table

The steps to determine whether a table of values determine a linear function are:

• Calculate the differences between every two consecutive \(x\) values.
• Calculate the differences between every two consecutive \(y\) values.
• Determine the ratio of the differences of \(y\) and differences of \(x\) for each pair of consecutive points.
• If the ratios are not equal, the function is not linear.

Here is an example table of values to consider:

Let us determine whether this table denotes a nonlinear function by using the steps mentioned above.

The function can be classified as a nonlinear function, given that the ratios of the differences of \(y\) to the differences of \(x\) are not uniform across the table of values.

Nonlinear Function Equation

A linear function can be represented mathematically in the form of \(f(x) = ax + b\), where ‘\(a\)’ and ‘\(b\)’ are constants. On the other hand, a nonlinear function is any function that does not satisfy the superposition principle, which means that it is not linear. 

Therefore, its equation can take any form other than the linear function equation of the form \(f(x) = ax + b\). 

Examples of nonlinear functions include polynomial functions, trigonometric functions, exponential functions, and logarithmic functions. It is important to note that the form of the equation of a nonlinear function may vary depending on the specific function involved.

• \(f(x) = x^{2}\) is nonlinear as it is a quadratic function.
• \(f(x) = 2^{x}\) is nonlinear as it is an exponential function.
• \(f(x) = x^{3} - 3x\) is nonlinear as it is a cubic function.

Nonlinear Graphs 

A nonlinear graph is a graph of a nonlinear function, which is a function that does not satisfy the superposition principle. This means that the function is not proportional to its input and does not have a constant rate of change. Nonlinear graphs can take many different shapes and forms, depending on the specific function involved.

Some examples of nonlinear graphs include:

• Parabolic graphs: These are graphs of quadratic functions, such as \(f(x) = x^{2}\). They have a U-shaped curve and are symmetric about the vertex.

Exponential graphs: These are graphs of exponential functions, such as \(f(x) = 2^{x}\). They have a rapid growth rate that increases exponentially as \(x\) increases.

• Logarithmic graphs: These are graphs of logarithmic functions, such as \(f(x) = \log(x)\). They have a curve that approaches but never touches the \(x\)-axis.

• Trigonometric graphs: These are graphs of trigonometric functions, such as \(f(x) = \sin(x)\). They have a repeating wave pattern that oscillates between the maximum and minimum values.

Nonlinear graphs are important in many fields, such as physics, economics, and engineering, where nonlinear relationships are often encountered.

The dissimilarities between linear and nonlinear functions are presented here.

Tips and Tricks on Nonlinear Functions

• A function is considered nonlinear if its graph does not form a straight line.
• A function is classified as nonlinear if its equation does not follow the form of \(f(x) = ax + b\).
• The function \(z = ax + by\) can either be linear or nonlinear, depending on its specific form.
• Nonlinear functions include a wide range of types, such as rational functions, polynomial functions, exponential functions, logarithmic functions, and more.

Non Linear Functions Summary

• A nonlinear function is a function whose graph is not a straight line.
• Nonlinear functions can take many forms, including polynomial, rational, exponential, and logarithmic functions.
• Nonlinear functions may have more than one \(x\)-intercept or no \(x\)-intercept at all.
• The slope of a nonlinear function varies at different points along its curve.
• Nonlinear functions can be modeled and analyzed using calculus and other advanced mathematical techniques.
• Nonlinear functions can have important applications in fields such as physics, engineering, economics, and biology.
• Nonlinear functions can be used to model complex real-world phenomena, such as population growth, disease spread, and financial markets.

Properties of Nonlinear Functions

1. Non-constant Rate of Change
   In a nonlinear function, the rate at which the output changes is not the same for equal changes in input. This means the slope keeps changing.
2. Graph is Not a Straight Line
   The graph of a nonlinear function is curved or has a shape that is not a straight line. It could be U-shaped, wavy, or grow/shrink sharply.
3. Can Have Many Shapes
   Nonlinear functions include many types such as:
   • Quadratic functions (parabolas)
   • Exponential functions (rapid growth or decay)
   • Trigonometric functions (like sine and cosine)
   • Logarithmic functions
4. Can Cross the X-axis More Than Once
   Nonlinear functions can have multiple x-intercepts, unlike linear functions which have at most one.
5. Can Have Turning Points
   These functions often have turning points, where the graph changes direction (like from increasing to decreasing).
6. Not Additive or Proportional
   In a nonlinear function, doubling the input doesn’t mean the output doubles. There is no simple multiplication or addition pattern.
7. May or May Not Be Continuous
   Some nonlinear functions are continuous (no breaks), while others can have jumps, holes, or undefined values.

Non Linear Functions Solved Examples

1.Which of the following functions are non-linear?

(a) f(x) = 7
(b) f(x) = 5x – 3
(c) f(x) = cos(x)

Solution:

• (a) f(x) = 7
  This is a constant function. It can also be written as f(x) = 0·x + 7, which is in the form of a linear equation.
  So, this is a linear function.
• (b) f(x) = 5x – 3
  This is in the form f(x) = ax + b, which is a straight line.
  So, this is also a linear function.
• (c) f(x) = cos(x)
  This is a trigonometric function, and its graph is a wave.
  So, this is a non-linear function.

Only (c) is a non-linear function.

2.Which of the following graphs represents non linear functions?

Solution:

By examining the graphs, we can deduce that functions that are nonlinear do not display straight lines but instead display curved lines. 

Therefore, based on the graphs provided, it can be concluded that graphs (a), (b), and (c) depict nonlinear functions, while graph (d) represents a linear function.

3.The following table shows the bank balances of Joe and Mitchell for the last \(5\) years. Graph the data and check if there has been any consistent growth for both of them.

Solution:

We will plot the points for both Joe and Mitchell.

Joe's points are displayed on a straight line, while Mitchell's points are on a curved line, both with positive slopes. 

It is evident that Joe's graph maintains a constant growth rate with a consistent rate of change of $\(100\), while Mitchell's graph portrays an inconsistent growth pattern with a curve. 

These observations indicate that Joe's growth rate has been constant over the past five years.

4. Does the following table represent a non-linear function?

Solution:

• The x-values increase by 1 each time.
• The y-values are being divided by 2 each time (100 → 50 → 25 → 12.5 → 6.25).

This is not a constant addition or subtraction.
Instead, it's a multiplicative pattern (getting halved each time).

So, the change is not constant, which means the function is not linear.

The function is non-linear.

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