Reciprocal Functions

Graph and analyze reciprocal functions y = 1/x. Grade 11 Math 20-1.

Lesson 6.7 of Rational Expressions and Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Reciprocal of a Linear Function. Explore what happens when you flip a linear function, discover vertical and horizontal asymptotes, and identify invariant points where the original and reciprocal share the same value.
• Reciprocal of a Quadratic Function. Extend reciprocal thinking to quadratics — find asymptotes using the discriminant, locate invariant points algebraically, and trace how the number of branches changes.
• Working Backwards from a Reciprocal. Given features of a reciprocal graph, recover the original function and sketch both curves — connecting asymptotes back to x-intercepts and flipping y-values.
• Designing Reciprocal Functions. Build a reciprocal function from a list of properties and determine whether two reciprocal graphs intersect — then prove an unknown coefficient using the discriminant.

Inside the lesson — a free preview

What Is a Reciprocal Function?

Definition. A reciprocal function is formed by replacing y with (1)/(y) in any function y = f(x). The result is

y = (1)/(f(x))

Examples:

The reciprocal of y = 2x is y = (1)/(2x).

The reciprocal of y = x^2 - 1 is y = (1)/(x^2 - 1).

Building Intuition with y = 2x. Compare y = 2x to its reciprocal y = (1)/(2x). Notice what flipping the y-value does:

Large becomes small. At x = 4, the original gives y = 8; the reciprocal gives y = (1)/(8).

Small becomes large. At x = (1)/(4), the original gives y = (1)/(2); the reciprocal gives y = 2.

Zero becomes undefined. At x = 0, the original gives y = 0; the reciprocal is undefined — you cannot divide by zero.

Sign is preserved. Where the original is positive, the reciprocal is positive; where negative, negative.

Key Features of a Reciprocal

Vertical Asymptote (VA). A vertical asymptote is a vertical line that the graph approaches but never touches.

For y = (1)/(f(x)), a VA occurs at every x-value where f(x) = 0, because division by zero is undefined.

Quick rule: Set f(x) = 0 and solve for x — each solution gives one VA.

Horizontal Asymptote (HA). A horizontal asymptote is a horizontal line that the graph approaches as x → ±∞.

For the reciprocal of any polynomial, the horizontal asymptote is always

y = 0 (the x-axis)

because as |f(x)| grows large, (1)/(f(x)) approaches zero.

Invariant Points. Invariant points are points that appear on both y = f(x) and y = (1)/(f(x)).

They occur where (1)/(y) = y, which is only possible when y = 1 or y = -1.

Quick rule: Set f(x) = 1 and solve for x — and then set f(x) = -1 and solve for x. Each solution is the x-coordinate of an invariant point.

Worked example: Asymptotes of Reciprocals

For each function, write the reciprocal and state the vertical and horizontal asymptotes.

Part a) — y = 4 - 3x. Step 1 — Write the reciprocal.

y = (1)/(4 - 3x)

Step 2 — Find the VA: set 4 - 3x = 0.

4 - 3x = 0 → 3x = 4 → x = (4)/(3)

Answer: VA: x = (4)/(3); HA: y = 0.

Part b) — y = 2x - 8. Step 1 — Write the reciprocal.

y = (1)/(2x - 8)

Step 2 — Find the VA: set 2x - 8 = 0.

2x = 8 → x = 4

Answer: VA: x = 4; HA: y = 0.

Part c) — y = x^2 - 3x - 10. Step 1 — Write the reciprocal.

y = (1)/(x^2 - 3x - 10)

Step 2 — Factor the denominator.

Two numbers multiplying to -10 and adding to -3: -5 and 2.

x^2 - 3x - 10 = (x - 5)(x + 2)

Step 3 — Set each factor equal to zero.

(x - 5) = 0 → x = 5

(x + 2) = 0 → x = -2

Answer: VAs: x = 5 and x = -2; HA: y = 0.

Part d) — y = 3x^2 + 2x - 8. Step 1 — Write the reciprocal.

y = (1)/(3x^2 + 2x - 8)

Step 2 — Factor the denominator.

Find two numbers multiplying to (3)(-8) = -24 and adding to 2: that is 6 and -4.

3x^2 + 2x - 8 = 3x^2 + 6x - 4x - 8

= 3x(x + 2) - 4(x + 2)

= (3x - 4)(x + 2)

Step 3 — Set each factor equal to zero.

(3x - 4) = 0 → x = (4)/(3)

(x + 2) = 0 → x = -2

Answer: VAs: x = (4)/(3) and x = -2; HA: y = 0.

Part e) — y = x^2 - 16. Step 1 — Write the reciprocal.

y = (1)/(x^2 - 16)

Step 2 — Factor as a difference of squares.

x^2 - 16 = (x - 4)(x + 4)

Step 3 — Set each factor equal to zero.

(x - 4) = 0 → x = 4

(x + 4) = 0 → x = -4

Answer: VAs: x = 4 and x = -4; HA: y = 0.

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