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 at which the function will actually be evaluated.

For example, 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 $2\pi$. This means $\sin(x)=\sin(x+2\pi)$ for all values of x.

However, the sine function has a period of $2\pi$ with respect to what? The sine function has a period of $2\pi$ with respect to its argument. The value of sine begins to repeat any time sine’s argument varies by $2\pi$, as illustrated below.

Figure 3.14.1 displays the graph of $y = \sin(3x)$. While it is indeed the case that $\sin(3x)=\sin(3x+2\pi)$, the period of $\sin(3x)$ with respect to x is not $2\pi$. Rather, sine's period with respect to 3x is $2\pi$. Whenever the value of 3x varies by $2\pi$, the value of $\sin(3x)$ repeats itself. However, for the value of $3x$ to vary by $2\pi$, the value of x need only vary by $\frac{2}{3}\pi$.

Figure 3.14.1. The period of $\sin(3x)$ with respect to $3x$ is $2\pi$. The period of $\sin(3x)$ with respect to x is $(2/3)\pi$.

So $\sin(3x)$ repeats itself whenever the value of $3x$ varies by $2\pi$. How much must the value of x vary so that the value of $3x$ varies by $2\pi$?

The value of $3x$ will vary by $2\pi$ whenever the value of x varies by $\frac{2}{3}\pi$: $$\mathrm{sin} \left(3\left(x+\frac{2}{3}π \right)\right)=\sin\left(3x+3\left(\frac{2}{3}\pi\right)\right)=\sin\left(3x+2\pi\right)$$

Therefore, the value of $\sin(3x)$ is periodic with respect to x, but with a period of $(2/3)\pi$. Whenever the value of x varies by $(2/3)\pi$, the value of $3x$ varies by $2\pi$, and therefore the value of $\sin(3x)$ repeats itself whenever the value of x varies by $(2/3)\pi$.

We can express the above reasoning formally. The value of $\sin(ax+b)$ repeats itself whenever $ax+b$ varies by $2\pi$. In other words, if we let $\Delta x$ represent the variation in the value of x that results in the value of $ax+b$ varying by $2\pi$, then $a(x+\Delta x)+b=(ax+b)+2\pi$. Solving for $\Delta x$, we get:

$$\begin{align}a(x+\Delta x)+b&=(ax+b)+2\pi\\[1ex] ax+a\Delta x+b&=(ax+b)+2\pi\\[1ex] (ax+b)+a\Delta x&=(ax+b)+2\pi\\[1ex] a\Delta x&=2\pi\\[1ex] \Delta x&=\frac{2\pi}{a}\end{align}$$

Therefore, the period of $\sin(ax+b)$ with respect to x is $\dfrac{2\pi}{a}$. Whenever the value of x varies by $2\pi/a$, the value of $ax+b$ varies by $2\pi$.

Exercise Set 3.14

1. The periods of sine and cosine with respect to their arguments is $2\pi$. 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.
1. $\sin\left(\dfrac{x}{a}+b\right)$
2. $\cos(mx-b)$
3. $\cos(4\sin(2x))$ Consider when the argument to $\cos()$ repeats itself and take into account that $\cos(-u)=\cos(u)$.
4. $2^{\sin(3x)}$
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.




1. Download this GC file and complete the assigned task.
2. Download this GC file and complete the assigned task.
3. Download this GC file and complete the assigned task.
4. Download this GC file and complete the assigned task.
5. Download this GC file and complete the assigned task.
6. Download this GC file and complete the assigned task.
7. 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 $\tan(x+\pi)=\tan(x)$ for all values of x at which $\tan(x)$ is defined. What is the period of $\tan(ax+b)$? Check your answer in GC.