CONSTANT FUNCTION: CHARACTERISTICS, EXAMPLES, EXERCISES - MATHS - 2025

The constant function is one in which the value of y is kept constant. In other words: a constant function always has the form f (x) = k, where k is a real number.

When graphing the constant function in the xy coordinate system, a straight line parallel to the horizontal or x-axis always results.

Figure 1. Graph of several constant functions on the Cartesian plane. Source: Wikimedia Commons. User: HiTe

This function is a particular case of the affine function, whose graph is also a straight line, but with a slope. The constant function has zero slope, that is, it is a horizontal line, as can be seen in figure 1.

There the graph of three constant functions is shown:

All are lines parallel to the horizontal axis, the first one is below said axis, while the rest are above.

Constant function characteristics

We can summarize the main characteristics of the constant function as follows:

-Its graph is a horizontal straight line.

-It has a unique intersection with the y axis, which is worth k.

-It's continuous.

-The domain of the constant function (the set of values that can have x) is the set of real numbers R.

-The path, range or counter-domain (the set of values ​​that the variable y takes) is simply the constant k.

Examples

Functions are necessary to establish links between quantities that depend on each other in some way. The relationship between them can be mathematically modeled, to find out how one of them behaves when the other varies.

This helps to build models for many situations and make predictions about their behavior and evolution.

Despite its apparent simplicity, the constant function has many applications. For example, when it comes to studying quantities that remain constant over time, or at least for an appreciable time.

In this way, magnitudes behave in situations such as the following:

-The cruising speed of a car moving along a long straight highway. As long as you do not brake or accelerate, the car has a uniform rectilinear motion.

Figure 2. If the car does not brake or accelerate, it has a uniform rectilinear motion. Source: Pixabay.

-A fully charged capacitor disconnected from a circuit has a constant charge over time.

-Finally, a flat rate parking lot maintains a constant price no matter how long a car is parked there.

Another way to represent a constant function

The constant function can alternatively be represented as follows:

Since any value of x raised to 0 gives 1 as a result, the previous expression reduces to the already familiar one:

Of course that happens as long as the value of k is different from 0.

That is why the constant function is also classified as a polynomial function of degree 0, since the exponent of the variable x is 0.

Solved exercises

- Exercise 1

Answer the following questions:

a) Can it be stated that the line given by x = 4 is a constant function? Give reasons for your answer.

b) Can a constant function have an x-intercept?

c) Is the function f (x) = w ² constant ?

Answer to

Here is the graph of the line x = 4:

Figure 3. Graph of the line x = 4. Source: F. Zapata.

The line x = 4 is not a function; by definition a function is a relation such that each value of the variable x corresponds to a single value of y. And in this case this is not true, since the value x = 4 is associated with infinite values ​​of y. Therefore the answer is no.

Answer b

In general, a constant function has no x-intercept, unless it is y = 0, in which case it is the x-axis itself.

Answer c

Yes, since w is constant, its square is also constant. What matters is that w does not depend on the input variable x.

- Exercise 2

Find the intersection between the functions f (x) = 5 and g (x) = 5x - 2

Solution

To find the intersection between these two functions, they can be respectively rewritten as:

They are equalized, obtaining:

What is a linear equation of the first degree, whose solution is:

The point of intersection is (7/5,5).

- Exercise 3

Show that the derivative of a constant function is 0.

Solution

From the definition of derivative we have:

Substituting in the definition:

Furthermore, if we think of the derivative as the rate of change dy / dx, the constant function does not undergo any change, therefore its derivative is zero.

- Exercise 4

Find the indefinite integral of f (x) = k.

Solution

Figure 4. Graph of the function v (t) for the mobile of exercise 6. Source: F. Zapata.

It asks:

a) Write an expression for the velocity function as a function of time v (t).

b) Find the distance traveled by the mobile in the time interval between 0 and 9 seconds.

Solution to

The graph shown shows that:

- v = 2 m / s in the time interval between 0 and 3 seconds

-The mobile is stopped between 3 and 5 seconds, since in this interval the speed is 0.

- v = - 3 m / s between 5 and 9 seconds.

It is an example of a piecewise function, or piecewise function, which in turn is composed of constant functions, valid only for the indicated time intervals. It is concluded that the desired function is:

Solution b

From the v (t) graph, the distance traveled by the mobile can be calculated, which is numerically equivalent to the area under / on the curve. In this way:

-Distance traveled between 0 and 3 seconds = 2 m / s. 3 s = 6 m

- Between 3 and 5 seconds he was detained, therefore he did not travel any distance.

-Distance traveled between 5 and 9 seconds = 3 m / s. 4 s = 12 m

In total the mobile traveled 18 m. Note that although the speed is negative in the interval between 5 and 9 seconds, the distance traveled is positive. What happens is that during that time interval, the mobile had changed the sense of its speed.

References

1. Geogebra. Constant functions. Recovered from: geogebra.org.
2. Maplesoft. The Constant Function. Recovered from: maplesoft.com.
3. Wikibooks. Calculation in a variable / Functions / Constant function. Recovered from: es.wikibooks.org.
4. Wikipedia. Constant function. Recovered from: en.wikipedia.org
5. Wikipedia. Constant function. Recovered from: es.wikipedia.org.