Mini-Lesson
Functions
This HL mini-lesson deepens Functions: polynomial division , the factor and remainder theorems , sum and product of roots , rational functions with oblique asymptotes, the modulus function , and odd/even symmetry.
Answer as you go and collect ⭐ stars. Press Start when ready.
Polynomials
Polynomials & division
A polynomial of degree n has the form aₙxⁿ + … + a₁x + a₀. Dividing P(x) by (x − a) gives a quotient and a remainder.
P(x) = (x − a)·Q(x) + Rwhere R is a constant when the divisor is linear
Quick check
Remainder theorem
? The remainder when P(x) is divided by (x − a) is:
P(a) ✅
P(−a) ❌
P(0) ❌
a ❌
Factor & remainder
The factor & remainder theorems
The remainder theorem : the remainder on dividing P(x) by (x − a) is P(a). The factor theorem : (x − a) is a factor exactly when P(a) = 0.
Worked example
P(x) = x³ − 2x² + 3x − 4 divided by (x − 2): remainder = P(2) = 8 − 8 + 6 − 4 = 2.
Calculate
Remainder theorem
✎ Find the remainder when x³ − 2x² + 3x − 4 is divided by (x − 2).
Check ✓
Hint: remainder = P(2) = 8 − 8 + 6 − 4.
Roots
Sum & product of roots
For ax² + bx + c = 0 with roots α, β:
α + β = −b/a · αβ = c/ageneralises to higher-degree polynomials (Vieta)
Quick check
Oblique asymptote
? A rational function has an oblique (slant) asymptote when the numerator degree is:
exactly one more than the denominator ✅
equal to the denominator ❌
less than the denominator ❌
two more than the denominator ❌
Calculate
Factor theorem
✎ Evaluate P(1) for P(x) = x³ − 6x² + 11x − 6 (to test whether x − 1 is a factor).
Check ✓
Hint: 1 − 6 + 11 − 6.
Rational
Rational functions & asymptotes
Rational functions have vertical asymptotes where the denominator is zero. Compare degrees for the end behaviour:
numerator degree < denominator: horizontal asymptote y = 0; degrees equal: horizontal asymptote y = ratio of leading coefficients; numerator degree one more: an oblique (slant) asymptote (found by division).
Sort it
Odd, even or neither?
Tap a function, then its symmetry type.
Modulus
The modulus function
|x| gives the distance of x from 0. The graph of y = |f(x)| reflects any part below the x-axis upward.
Worked example
Solve |x − 3| = 4: either x − 3 = 4 (x = 7) or x − 3 = −4 (x = −1).
Calculate
Sum of roots
✎ Find the sum of the roots of 2x² − 5x + 3 = 0.
Check ✓
Hint: sum of roots = −b/a = −(−5)/2.
Quick check
Even function
? An even function satisfies f(−x) = f(x). Its graph is symmetric about:
the y-axis ✅
the x-axis ❌
the origin ❌
the line y = x ❌
Odd & even
Odd & even functions
An even function has f(−x) = f(x) (symmetry about the y-axis); an odd function has f(−x) = −f(x) (rotational symmetry about the origin).
x², cos x and |x| are even; x³, sin x and 1/x are odd; most functions are neither.
Calculate
Product of roots
✎ Find the product of the roots of 2x² − 5x + 3 = 0.
Check ✓
Hint: product of roots = c/a = 3/2.
Match it
Match each fact to its result
Tap a statement on the left, then its matching partner on the right.
Self-inverse
Composite transformations & self-inverse
Combine transformations carefully — inside changes act on x (and reverse), outside on y. A function is self-inverse when f(f(x)) = x.
Example: f(x) = 1/x satisfies f(f(x)) = x, so it is its own inverse; its graph is symmetric in y = x.
Calculate
Modulus equation
✎ Solve |x − 3| = 4 and give the larger solution.
Check ✓
Hint: x − 3 = 4 gives one solution; x − 3 = −4 the other.
Quick check
Self-inverse
? Which function is its own inverse (self-inverse)?
f(x) = 1/x ✅
f(x) = 2x ❌
f(x) = x + 1 ❌
f(x) = x² ❌
Strategy
Working at HL
Use the remainder theorem to test factors quickly, Vieta to link coefficients and roots, and degree comparison to find asymptotes without a graph.
Exam habit: a modulus equation usually has two cases — always check both solutions in the original equation.
Recap
The big ideas to know
Division: P(x) = (x−a)Q(x) + R
Theorems: remainder P(a); (x−a) a factor ⇔ P(a)=0
Vieta: α+β = −b/a, αβ = c/a
Asymptotes: compare degrees; oblique when numerator one higher
Symmetry: even f(−x)=f(x); odd f(−x)=−f(x); self-inverse f(f(x))=x
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