Functions and Their Graphs – Rational Functions Review

These flashcards cover key ideas about representing rational functions, determining domain, range, intercepts, zeros, and asymptotes, and applying these concepts to real-life contexts such as cost analysis and engineering constraints.

What three basic ways can a rational function be represented?

By a table of values, by its graph, and by its equation.

In the Pueblo por la Playa problem, what is the rational function that models the average cost per day for x days?

f(x)= (300,000 + 700x) / x

Why is f(0) undefined for the Pueblo por la Playa cost function?

Because division by zero is undefined; x (the number of days) is in the denominator.

What general form must a rational function follow?

f(x)= p(x)/q(x) where p(x) and q(x) are polynomials and q(x) ≠ 0.

State the three conditions that p(x) and q(x) must satisfy for f(x)=p(x)/q(x) to be a rational function.

1) p(x) and q(x) are polynomials with no negative or fractional exponents, 2) q(x) ≠ 0, 3) The domain excludes all x where q(x)=0.

Define the domain of a function.

The set of all x-values that the function can accept (input values).

Define the range of a function.

The set of all y-values (or f(x) values) that the function can output.

What are the three common notations used to write domain and range?

Roster form, set-builder notation, and interval notation.

How is the domain of f(x)=5/(x−3) written in interval notation?

(-∞,3) ∪ (3,∞)

In an ordered pair (x,y), what is the x-value called?

The abscissa.

In an ordered pair (x,y), what is the y-value called?

The ordinate.

In function notation, which variable is considered the dependent variable?

f(x) or y (because it depends on x).

How do you find the y-intercept of a rational function?

Substitute x = 0 into the function and compute f(0).

How do you find the x-intercept(s) of a rational function?

Set the numerator equal to zero, solve for x, and ensure those x-values do not make the denominator zero.

What is a zero of a rational function?

An x-value that makes the numerator zero without simultaneously making the denominator zero.

Why does f(x)=(x+4)(x+2)/(x−3)(x+2) have zero x=−4 but not x=−2?

x=−2 makes both numerator and denominator zero, so it is excluded; only x=−4 remains as a zero.

Define a vertical asymptote (VA).

A vertical line x = a where the function grows without bound as x approaches a, arising from values that make the denominator zero after reduction.

How do you locate vertical asymptotes in a rational function?

Factor and simplify the function, set the (reduced) denominator equal to zero, solve for x.

What is the VA of f(x)=1/(x+5)?

x = −5

Give the VAs of f(x)= (x+2)/[(x+1)(x−4)].

x = −1 and x = 4

State the rule for horizontal asymptotes when degree(numerator) < degree(denominator).

The horizontal asymptote is y = 0.

State the rule for horizontal asymptotes when degree(numerator) = degree(denominator).

The horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator).

State the rule for horizontal asymptotes when degree(numerator) > degree(denominator).

There is no horizontal asymptote; a slant/oblique asymptote may exist instead.

What is the horizontal asymptote of f(x)=(3x+8)/(2x+1)?

y = 3/2

Does f(x)=8x³/(x−2) have a horizontal asymptote?

No, because the numerator’s degree (3) is greater than the denominator’s degree (1).

When does a slant (oblique) asymptote occur?

When the numerator’s degree is exactly one more than the denominator’s degree.

How do you find a slant asymptote?

Perform polynomial long (or synthetic) division; the quotient (without the remainder) gives the slant line y = mx + b.

What is the slant asymptote of h(x)=(x²+3)/(x−1)?

y = x + 1

Explain why the average cost per day at Pueblo por la Playa decreases as the number of days increases.

Because the fixed membership fee (300,000) is spread over more days, lowering the average cost, while the daily fee remains constant at 700 pesos.

List the daily average costs for staying 5, 10, 15, 20, 25, and 30 days at Pueblo por la Playa.

5 days: ₱60,700; 10 days: ₱30,700; 15 days: ₱20,700; 20 days: ₱15,700; 25 days: ₱12,700; 30 days: ₱10,700.

Why does the graph of the Pueblo por la Playa cost function form a curve instead of a straight line?

Because the function is rational (division by x) and not linear; the rate of change is not constant.

What physical concept is commonly linked to vertical asymptotes in engineering?

Physical constraints where a quantity becomes impossible or infinite, such as infinite tension on a bridge cable at a certain load.

In rational functions, what causes your calculator or computer to output “undefined” or an error?

Attempting to evaluate the function at an x-value that makes the denominator zero.

For f(x)=1/x, what are the domain and range?

Domain: all real numbers except 0; Range: all real numbers except 0.

How would you write the domain of g(x)=2/(x+1) in set-builder notation?

{ x | x ∈ ℝ, x ≠ −1 }

If a rational function reduces so that a factor cancels between numerator and denominator, what graphical feature often results at that x-value instead of a vertical asymptote?

A hole (removable discontinuity) in the graph.

Why might a rational function have no zeros even though its numerator appears to have roots?

Because every root of the numerator also makes the denominator zero, and such values are excluded from the domain.

What are the learning targets related to intercepts and asymptotes in the lecture?

Identify x- and y-intercepts and zeros; determine vertical and horizontal asymptotes; relate asymptotic behavior to real-world constraints.