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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 to get a value of the function. |

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

However, the sine function is periodic by $2\pi$ 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.

x (radians) |
$\sin(3x)$ | $\sin(3x+2\pi)$ |

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 $y = \sin(3x)$. 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 3 x__

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 $2/3\pi$:
$$\mathrm{sin} \left(3\left(x+\frac{2}{3}π \right)\right)=\sin\left(3x+3\left(\frac{2}{3}\pi\right)\right)=\sin\left(3x+2\pi\right)$$

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 $2\pi$.

- 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. - $\sin\left(\dfrac{x}{a}+b\right)$
- $\cos(mx-b)$
- $\cos(4\sin(2x))$
*Consider when the argument to $\cos()$ repeats itself and take into account that $\cos(-u)=\cos(u)$.* - $2^{\sin(3x)}$
- 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. - Download this GC file and complete the assigned task.
- Download this GC file and complete the assigned task.
- Download this GC file and complete the assigned task.
- Download this GC file and complete the assigned task.
- Download this GC file and complete the assigned task.
- Download this GC file and complete the assigned task.
- Download this GC file and complete the assigned task.
- 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.

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