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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

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 $y = f(x)$,

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 (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(3

So sin(3

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

Therefore, the value of sin(3

We can express the above reasoning formally. The value of sin(

a) sin($\frac{x}{a} + b$)

b) cos(

c) cos(4sin(2

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(