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Section 3.14
Independent Variable vs. Argument of Function


When Euler introduced modern function notation in 1734, he stated, “If $f(\frac {x}{a} + c)$ denotes any function of $ \frac {x}{a} + c$ …”. His use of parentheses introduced a distinction that we describe today as the distinction between a function’s independent variable and a function's argument.

A function’s argument is the expression that represents the value that will actually be passed to the function’s definition for evaluation.

So, in $y = f(x)$, x is f’s independent variable and x is f’s argument. However, in $y = f(\frac{x}{3} + 5)$, x is f’s independent variable while $\frac{x}{3} + 5$ is f’s argument.

The distinction between independent variable and argument can be very useful. You know that the sine function is periodic with period 2π. This means that sin(x) = sin(x+2π) for all values of x. However, the sine function is periodic with respect to what? The sine function has a period of 2π with respect to its argument. The value of sine begins to repeat any time sine’s argument changes by 2π, as illustrated below.

x (radians) sin(3x) sin(3x + 2π)
0.25 0.6816 0.6816
0.4 0.9320 0.9320
1.7 0.9258 0.9258


Figure 3.14.1 displays the graph of $y = \sin(3x)$. The period of sin(3x) with respect to x is not 2π. But sine's period with respect to 3x is 2π. Whenever the value of 3x changes by 2π, the value of sin(3 x) repeats itself.
period of sin(3x)
Figure 3.14.1. The period of sin(3x) with respect to 3x is 2π. The period of sin(3x) with respect to x is (2/3)π.

So sin(3x) repeats itself whenever the value of 3x changes by 2π. But how much must the value of x change so that the value of 3 x changes by 2π? The value of 3x will change by 2π whenever the value of x changes by (2/3)π:.
$$\mathrm{sin} \left( 3 \left( x+\frac{2}{3}π \right) \right) = \mathrm{sin} \left( 3x + 3 \left( \frac{2}{3}π \right) \right) = \mathrm{sin} \left( 3x + 2π \right) $$
Therefore, the value of sin(3x) is periodic with respect to x, but with a period of (2/3)π. Whenever the value of x changes by (2/3)π, the value of 3x changes by 2π, and therefore the value of sin(3 x) will repeat itself whenever the value of x changes by (2/3)π.

We can express the above reasoning formally. The value of sin(ax + b) repeats itself whenever $ax + b$ changes by 2π. In other words, if we let ∆x represent the change in the value of x that results in the value of $ax + b$ changing by 2π, then a(x + ∆x) + b = (ax + b) + 2π. Solving for ∆x, we get:

solving for delta x
Therefore, the period of sin(ax + b) with respect to x is $\frac{2π}{a}$.

Exercise Set 3.14

1. The periods of sine and cosine with respect to their arguments is 2π. Use the reasoning illustrated in this section to determine the period of each of the following with respect to x. Explain your reasoning. Check your answers in GC by assigning values to the statements’ parameters.

a) sin($\frac{x}{a} + b$)

b) cos(mx - b)

c) cos(4sin(2x))

d) $2^{\cos(3x)}$

2. The animation below illustrates entering an argument for a function so that GC’s display of its graph passes through the given points. Do the same for 2.a-2.e. Exercises 2.f and 2.g ask you to do the reverse of a-e. They ask you to define the function f so that GC’s graph of y = f(x) passes through the given points.


a) Download this GC file and complete the assigned task.

b) Download this GC file and complete the assigned task.

c) Download this GC file and complete the assigned task.

d) Download this GC file and complete the assigned task.

e) Download this GC file and complete the assigned task.

f) Download this GC file and complete the assigned task.

g) Download this GC file and complete the assigned task.

3. The tangent function has a period of π with respect to its argument. This means that tan(x + π) = tan(x) for all values of x at which tan(x) is defined. What is the period of tan( ax + b)? Check your answer in GC.