Reading the Draft: Functions (Introduction)

Flashcards covering definitions of functions, notation, domain, range, vertical line test, and basic evaluation and graph concepts from the lecture notes.

What is a function, as defined in the notes?

A function is a dependence of one quantity (output) on another quantity (input), where each input value has exactly one output value.

In function notation f(x), what do the letters x and f(x) represent?

x is the input/independent variable; f(x) is the output/dependent value.

What property must input values satisfy for a relation to be a function?

No input value can map to more than one output value.

Provide a real-life example of a function mentioned in the notes.

The temperature of coffee as a function of time since it was poured.

Provide another real-life example of a function from the notes.

Distance traveled as a function of speed (in a car).

Provide another real-life example of a function from the notes.

The cost of mailing a package as a function of its weight.

For the function f(x) = x^2 + 4, what is f(3)?

f(3) = 3^2 + 4 = 9 + 4 = 13.

When evaluating f(-2) for f(x) = x^2 + 4, what must you be careful about?

Use parentheses so that (-2)^2 is used, not -2^2; (-2)^2 = 4, so f(-2) = 8.

What is the domain of the function f(x) = x^2 + 4, as discussed in the notes?

All real numbers.

What is the Vertical Line Test used for?

To determine if a graph represents a function; a function’s graph must be intersected by any vertical line at most once.

What does the notes say about the range in the example graph?

There are no negative y-values, and y = 0 occurs at (0,0), so the range is y ≥ 0 (i.e., [0, ∞)).

If a domain excludes a single value, how is it written in interval notation?

As (-∞, a) ∪ (a, ∞); e.g., excluding 0: (-∞, 0) ∪ (0, ∞).

What does it mean to find x-values where f(x) = g(x) on a graph?

It means finding the x-values where f and g have the same y-value—the points where their graphs intersect.

Where are inputs and outputs typically represented on a graph of a function?

Inputs correspond to the x-values on the horizontal axis (domain); outputs correspond to the y-values (range).