Week 1 - Functions and Exponentials and Logarithms

What is a function according to Stewart's definition?

A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E.

What must be true for a function to be well-defined?

Well-defined functions must have a single output value for any given input value.

What is the domain of the function f(x) = 4/(x − 2)?

D = {x|x ̸= 2}, since the function is undefined at x = 2.

What is the range of the function f(x) = 4/(x − 2)?

R = {x|x ̸= 0}, because the output cannot be 0.

What rule is used to determine whether a graph represents a function?

The vertical line test.

What characterizes a piecewise function?

A piecewise function is defined by different formulas in different parts of its domain.

What does it mean for a function to be increasing on an interval I?

A function f is increasing on an interval I if f(x1) < f(x2) whenever x1 < x2 in I.

What is a polynomial function according to Stewart?

A function P is a polynomial if P(x) = anxn + an−1xn−1 + ... + a2x2 + a1x + a0 where n is a nonnegative integer.

How can you identify if a function is a power function?

A function f(x) is a power function if it can be expressed in the form f(x) = xa, where a is a constant.

What is the basic property of exponential functions?

If b > 0 and b ̸= 1, then f(x) = bx is a continuous function with domain R and range (0, ∞).

What does it mean for a function to be odd according to Stewart?

A function f is called odd if f(−x) = −f(x) for every number x in its domain.

What is the effect of adding a constant c to a function f(x)?

This results in a vertical translation of the graph: y = f(x) + c shifts the graph upward by c units.

What is the composition of functions?

The composite function f ◦ g is defined by (f ◦ g)(x) = f(g(x)).