Math 112 Precalculus Mathematics - Functions and Graphs (Flashcards)

A set of practice flashcards covering key concepts about functions, graphs, domain/range, intercepts, slope, and basic operations with equations from the lecture notes.

What does the vertical line test determine about a relation on a graph?

If every vertical line intersects the graph at most once, the relation is a function; if any vertical line intersects more than once, it is not a function.

Define the domain of a relation.

The set of all x-values for which the relation is defined.

Define the range of a relation.

The set of all possible y-values produced by the relation.

How can you tell from a table whether the relation is a function?

If no x-value repeats with a different y-value; each x maps to a single y.

What is an x-intercept?

The point(s) where the graph crosses the x-axis; y = 0.

What is a y-intercept?

The point where the graph crosses the y-axis; x = 0.

What is slope-intercept form?

The equation y = mx + b, where m is the slope and b is the y-intercept.

How do you compute the slope from two points (x1,y1) and (x2,y2)?

m = (y2 − y1) / (x2 − x1).

How do you obtain f(x) from a graph when evaluating at a specific x?

Read the y-value corresponding to the input x on the graph.

What is the domain of f(x) = sqrt(x)?

[0, ∞)

What is the domain of f(x) = sqrt(x^2 − 16)?

(-∞, −4] ∪ [4, ∞)

What is the domain of f(x) = sqrt(2x − 40)?

[20, ∞)

What is the discriminant and what does it tell you?

D = b^2 − 4ac; D > 0: two real roots; D = 0: one real root; D < 0: two complex roots.

What does (f ∘ g)(x) mean?

The composition f(g(x)).

If (x, y) is on the graph of h, what is h(x)?

h(x) = y.

What is the slope of the line through (2,3) and (3,6)?

m = (6 − 3) / (3 − 2) = 3.

Given the line 10x − 9y = −8, what are its slope and y-intercept?

Slope m = 10/9; y-intercept b = 8/9.

What is vertex form of a parabola and its vertex?

y = a(x − h)^2 + k with vertex at (h, k).

If a piecewise function is defined for all x, what is its domain?

All real numbers.

Find the equation of the line through (9,5) and (−10,4) in slope-intercept form.

y = (1/19)x + 86/19.

Solve for y in terms of x from −5x + y = −2.

y = 5x − 2.

What is the difference quotient f(x + h) − f(x) over h as h → 0?

It defines the derivative of f at x.

What is the domain of f(x) = sqrt(x + 2)?

x ≥ −2.

What are the x- and y-intercepts of the circle x^2 + y^2 = 64?

x-intercepts: (−8,0) and (8,0); y-intercepts: (0,−8) and (0,8).

Is f(n) = 1/(n^2 + 7) a function from Z to R?

Yes; every integer input n maps to a single real output.

How do you determine if a relation given as a set of ordered pairs is a function?

No x-value maps to two different y-values; if an x repeats with different y-values, it is not a function.

What is the domain of f(x) = 1/x?

All real numbers except x = 0.

What is the domain of f(x) = sqrt(-2x)?

x ≤ 0.

What is the x-intercept and y-intercept of the line 3x − 4y = −4?

x-intercept: (−4/3, 0); y-intercept: (0, 1).

Find the equation of a line through (6,2) with slope −1/4.

y = −(1/4)x + 7/2.

If 9x − 4y = 12, what are the slope and y-intercept?

Slope m = 9/4; y-intercept b = −3.

Solve the inequality (x−4)(x+6) < 0.

−6 < x < 4.

If f(x) = 8x^2 + x + 5 = 0, what is the sign of the discriminant?

D = b^2 − 4ac; here D < 0, so two complex roots.

What is the vertex of y = (x − 1/2)^2 − 8?

Vertex at (1/2, −8).