MATH-C2210 CHAPTER 1 - REVIEW

What is a function?

A type of relation that assigns every input, x, to exactly one output, f(x).

What is the domain and range of a function?

The domain is represented by a set of acceptable inputs and the range as the set of outputs.

How do you know a given curve is the graph of a function?

If the curve passes the vertical line test, in which no vertical line will overlap at two or more points.

Why does the vertical line test work?

The y-values in the range cannot be repeated, thus no overlapping on the graph.

What are the four methods of representing functions?

1. Verbal

2. Numeric

3. Visual

4. Algebraic

What is a piecewise function?

A function consisting of two or more subfunctions, defined at a specific interval over the domain.

What is an even function?

f(-x) = f(x). In other words, when the negative x-values produce a positive y-value or output.

What is the symmetry of an even function?

Symmetric with respect to the y-axis

What is an odd function?

f(-x) = - f(x). In other words, the negative x-values in the domain produce negative y-values or outputs.

What is the symmetry of an odd function?

Symmetric with respect to the origin, (0,0).

When is a function increasing on an interval?

When x1 < x2, f(x1) < f(x2). In other words, the x-values and y-values both increase at a certain interval.

When is a function decreasing on an interval?

When x₁ < x₂, f(x₁) > f(x₂). In other words, as the x-values increase, the y-values decrease.

What are the two functions with both a domain and range at ( - infinity, + infinity)?

1. Linear functions

2. Polynomial functions

What is the domain and range of a constant function, f(x) = b?

DOMAIN: ( - infinity, + infinity)

RANGE: {b}

What affects the domain and range of a power function?

The value of the exponent.

What are the three types of power functions with unique domains?

1. Polynomial functions

2. Reciprocal functions

3. Root functions

What is the domain and range of a power function, f(x) = x^a, where a is any positive integer (whole number)?

DOMAIN: ( - infinity, + infinity)

RANGE: ( - infinity, + infinity)

What is the domain and range of a reciprocal function, f(x) = x^a, where a is any negative integer (whole number)?

DOMAIN: ( - infinity, 0) U (0, infinity)

RANGE: ( - infinity, 0) U (0, infinity)

If a = 1/n, where n is a positive integer > or = 2, and n is odd; what is the domain and range of the root function f(x) = x^a?

DOMAIN: ( - infinity, + infinity)

RANGE: ( - infinity, + infinity)

If a = 1/n, where n is a positive integer > or = 2, and n is even; what is the domain and range of the root function f(x) = x^a?

DOMAIN: [0, + infinity)

RANGE: [0, + infinity)

What is the domain of a rational function f(x) = P(x) / Q(x)?

All real numbers, except the denominator cannot equal 0.

What is the domain and range of sinx and cosx?

DOMAIN: ( - infinity, + infinity)

RANGE: [-1,1]

What is the domain and range of tanx?

DOMAIN: All real numbers, but x cannot equal (2k + 1)pi / 2 where k is an integer

RANGE: ( - infinity, + infinity)

What is the domain and range of cotx?

DOMAIN: All real numbers, but x cannot equal (k * pi) / 2 where k is an integer

RANGE: ( - infinity, + infinity)

What is the domain and range of secx?

DOMAIN: All real numbers, but x cannot equal (2k + 1)pi / 2 where k is an integer

RANGE: ( - infinity, -1] U [1, + infinity)

What is the domain and range of cscx?

DOMAIN: All real numbers, but x cannot equal (k * pi) / 2 where k is an integer

RANGE: ( - infinity, -1] U [1, + infinity)

Which trig functions have a period length of 2pi?

1. sinx

2. cosx

3. cscx

4. secx

Which trig functions have a period length of pi?

1. tanx

2. cotx

What is the domain and range of the exponential function f(x) = a^x?

DOMAIN: ( - infinity, + infinity)

RANGE: (0, + infinity)

At which interval does f(x) = a^x increase on its domain?

When a > 1

At which interval does f(x) = a^x decrease on its domain?

When 0 < a < 1

What is the domain and range of f(x) = log_ax?

DOMAIN: (0, + infinity)

RANGE: ( - infinity, + infinity)

At which interval does f(x) = log_ax increase on its domain?

When a > 1

At which interval does f(x) = log_ax decrease on its domain?

When 0 < a < 1

When is a function said to be one-to-one?

f(x₁) cannot equal f(x₂) whenever x₁ cannot equal x_2. In other words, when no two distinct inputs correspond to the same output.

How do you know a given curve is the graph of a one-to-one function?

If the graph passes the horizontal line test, where no horizontal line intersects on the graph more than one point.

When are functions considered invertible?

When the function is one-to-one.

What is the rule for the domain and range of inverse functions?

The domain and range from the parent function are swapped at the inverse function.

What is the change-of-base logarithm property?

log_ax = lnx / lna

How are invertible trig functions produced?

The six trig functions are restricted at domains in which they are one-to-one.

What is the domain and range of invertible sinx? (in which sinx can become inverse)

DOMAIN: [ - pi/2, pi/2 ]

RANGE: [-1,1]

What is the domain and range of sin^-1(x)?

DOMAIN: [-1,1]

RANGE: [ - pi/2, pi/2 ]

What is the domain and range of invertible cosx? (in which cosx can become inverse)

DOMAIN: [0, pi]

RANGE: [-1,1]

What is the domain and range of cos^-1(x)?

DOMAIN: [-1,1]

RANGE: [0, pi]

What is the domain and range of invertible tan(x)? (in which tan(x) can become inverse)

DOMAIN: ( - pi/2, pi/2 )

RANGE: ( - infinity, + infinity)

What is the domain and range of tan^-1(x)?

DOMAIN: ( - infinity, + infinity)

RANGE: ( - pi/2, pi/2 )