D. Characteristics of Functions · Combining Functions and Modelling

Properties of Combined Functions: Even, Odd, Increasing, Decreasing

D2.3About 70 minutes

Learning goals

I can test a function algebraically to decide whether it is even, odd or neither.
I can predict and prove whether a sum, difference, product or quotient of even and odd functions is even, odd or neither.
I can explain when sums and products of increasing or decreasing functions are increasing or decreasing.
I can use these properties to sketch and check combined functions.

Words to know

even function
A function with f(−x) = f(x) for every x in its domain. Its graph is symmetric in the y-axis.

odd function
A function with f(−x) = −f(x) for every x in its domain. Its graph has rotational symmetry of 180° about the origin.

increasing on an interval
For any a < b in the interval, f(a) < f(b): the graph rises from left to right.

decreasing on an interval
For any a < b in the interval, f(a) > f(b): the graph falls from left to right.

counterexample
One specific example that shows a general statement is false.

Symmetry you can test

Fold a paper snowflake in half and both halves match. Turn the letter S or the number 69 upside down and it looks the same. Graphs can have both kinds of symmetry, and you can test for them with algebra instead of folding.

Remember from the sibling unit Polynomial Functions and Equations:

f is even if f(−x) = f(x). The graph is a mirror image in the y-axis. Examples: x², x⁴, |x|, cos x.
f is odd if f(−x) = −f(x). Turning the graph 180° about the origin leaves it unchanged. Examples: x, x³, 1/x, sin x, tan x.
Most functions are neither: for example 2ˣ, log x, x² + x.

The domain of an even or odd function must be symmetric too: if x is in it, so is −x. That is why log x (domain x > 0) can never be even or odd.

Even and odd: the combination rules

Suppose f and g are both even. Then

(f + g)(−x) = f(−x) + g(−x) = f(x) + g(x) = (f + g)(x), so f + g is even.

Suppose f and g are both odd. Then

(f + g)(−x) = f(−x) + g(−x) = −f(x) − g(x) = −(f + g)(x), so f + g is odd.

For products, both odd gives

(fg)(−x) = f(−x) g(−x) = (−f(x))(−g(x)) = f(x) g(x), so fg is even.

The same kind of reasoning gives the whole table below. A handy memory trick for products and quotients: think of even as +1 and odd as −1 and multiply. (Odd × odd is like (−1)(−1) = +1, so even.) There is no such trick for sums: an even function plus an odd function is usually neither.

f	g	f + g and f − g	fg and f/g
even	even	even	even
odd	odd	odd	even
even	odd	usually neither	odd

Rules for combining even and odd functions (on a domain where the combination is defined and symmetric). "Usually neither": the exception is when one of the two is the zero function.

Left: an even + even function is symmetric in the y-axis; the points at a and −a have the same height. Right: an odd + odd function is unchanged by a half-turn about the origin; the points at a and −a have opposite heights.

Worked example

Example 1: Classify each function algebraically

Decide whether each is even, odd or neither: (a) h(x) = x sin x (b) k(x) = x³ cos x (c) m(x) = 2ˣ + 2⁻ˣ (d) n(x) = x² + sin x.

(a) h(−x) = (−x) sin(−x) = (−x)(−sin x) = x sin x = h(x). Even (odd × odd). This is why h(π/2) = h(−π/2) in the lesson Products of Functions.
(b) k(−x) = (−x)³ cos(−x) = (−x³)(cos x) = −x³ cos x = −k(x). Odd (odd × even).
(c) m(−x) = 2⁻ˣ + 2⁻⁽⁻ˣ⁾ = 2⁻ˣ + 2ˣ = m(x). Even, even though 2ˣ and 2⁻ˣ are each neither.
(d) n(−x) = x² − sin x. Compare with n(x) = x² + sin x and −n(x) = −x² − sin x. Test x = π/2: n(π/2) = π²/4 + 1 ≈ 3.47 and n(−π/2) = π²/4 − 1 ≈ 1.47. Neither equal nor opposite, so neither.

(a) even (b) odd (c) even (d) neither. For "neither", one numerical counterexample is a complete proof.

How to show "neither"

To prove a function is even or odd you need algebra that works for every x. To prove it is neither, one counterexample is enough: find one value a with f(−a) ≠ f(a) and f(−a) ≠ −f(a).

Increasing and decreasing

A function is increasing on an interval if bigger inputs always give bigger outputs there: a < b implies f(a) < f(b). (In the sibling unit Rates of Change this matches a positive rate of change.)

Sums. If f and g are both increasing on an interval, so is f + g. Proof: if a < b, then f(a) < f(b) and g(a) < g(b). Adding the two inequalities gives f(a) + g(a) < f(b) + g(b).

Differences. If f is increasing and g is decreasing, then −g is increasing, so f − g = f + (−g) is increasing. But the difference of two increasing functions can do anything: x − x/2 = x/2 increases, while x − 2x = −x decreases.

Products. If f and g are both increasing and both positive on an interval, then fg is increasing there: for a < b, f(a)g(a) < f(b)g(a) < f(b)g(b) (each step multiplies by a positive number). Without positivity this fails: f(x) = x and g(x) = x are increasing, but their product x² is decreasing for x < 0.

Reciprocals and quotients. If f is positive and increasing, then 1/f is decreasing (a bigger denominator makes a smaller fraction). So if f is positive and increasing and g is positive and decreasing, f/g is increasing.

Left: 2ˣ and x are both increasing, so their sum is increasing everywhere. Right: x and x are both increasing, but their product x² decreases on the shaded part, x < 0, where the factors are negative.

Worked example

Example 2: Using increasing functions to count solutions

Show that the equation 2ˣ + log x = 5 has exactly one solution, and find which whole numbers it lies between.

Let h(x) = 2ˣ + log x, with domain x > 0 (because of log x).
2ˣ is increasing and log x is increasing on x > 0, so their sum h is increasing on x > 0.
An increasing function can take the value 5 at most once: if h(a) = h(b) = 5 with a < b, then h(a) < h(b), a contradiction.
Check values: h(2) = 4 + log 2 ≈ 4.30 < 5 and h(3) = 8 + log 3 ≈ 8.48 > 5.
Since h moves continuously from below 5 to above 5, it equals 5 somewhere between x = 2 and x = 3.

Exactly one solution, between x = 2 and x = 3. (You will find it to more decimal places in the lesson When Algebra Is Not Enough: Solving Graphically and Numerically; it is about 2.22.)

Worked example

Example 3: Predicting properties before graphing

Without graphing, predict the symmetry of y = (sin x)/x and of y = x² · 2ˣ, and whether y = x · 2ˣ is increasing on x > 0.

(sin x)/x is odd ÷ odd, so it is even. This matches its graph in the lesson Quotients of Functions, which is symmetric in the y-axis.
x² · 2ˣ is even × neither. Test x = 1: 1 · 2 = 2; x = −1: 1 · 0.5 = 0.5. Not equal and not opposite: neither.
On x > 0, both x and 2ˣ are positive and increasing, so x · 2ˣ is increasing on x > 0.

(sin x)/x is even; x² · 2ˣ is neither; x · 2ˣ is increasing for x > 0.

Just for fun

Why is cos x always so calm, even in an exam?

It is even-tempered. (Its neighbour sin x is a bit odd, but very well rounded.)

Hands-on activity

Fold-and-turn symmetry test

You need: two sheets of grid paper (or lined paper with axes drawn carefully), a pencil and a calculator in radian mode.

On sheet 1, draw axes for −3 ≤ x ≤ 3 and plot h(x) = x³ − 3x using x = −2, −1.5, −1, −0.5, 0, 0.5, 1, 1.5, 2. Join the points smoothly.
On sheet 2, plot k(x) = x² cos x for the same x-values.
Fold test: fold each sheet along the y-axis and hold it up to a window. Does the graph match itself?
Turn test: turn each sheet upside down (a half-turn). Does the graph look exactly the same as before?
Predict from the rules first: x³ − 3x is odd − odd; x² cos x is even × even.

What to look for: h(x) = x³ − 3x passes the turn test (odd): for example h(1) = −2 and h(−1) = 2. k(x) = x² cos x passes the fold test (even): for example k(1) ≈ 0.540 and k(−1) ≈ 0.540. Neither passes both tests.

Your turn

Is f(x) = x⁴ − 3x² + 1 even, odd or neither? Show the algebra.

Is g(x) = x³ + sin x even, odd or neither?

Is h(x) = x + cos x even, odd or neither? Give a counterexample.

Predict the symmetry of y = tan x · sin x.

f and g are both increasing on 0 < x < 5. Must f + g be increasing there? Must f − g?

Is y = 3ˣ + x³ increasing everywhere? Explain.

Is y = x²/(x² + 1) even, odd or neither?

More practice: warm-up

Name one even trigonometric function and one odd one.

Odd × even = ?

Odd + odd = ?

If f is even and f(3) = 8, what is f(−3)?

If g is odd and g(2) = −5, what is g(−2)?

More practice: on your own

Show algebraically that y = 3ˣ − 3⁻ˣ is odd.

Classify y = x sin x + cos x and y = x cos x + sin x.

f is odd and defined at 0. Prove that f(0) = 0.

Is y = 1/(1 + 2ˣ) increasing or decreasing? Explain without graphing.

Give an example of two decreasing functions whose product is increasing on some interval.

Challenge

Prove that every function f with a symmetric domain is the sum of an even function and an odd function. (Hint: try E(x) = [f(x) + f(−x)]/2.)

Can a function be both even and odd? Explain.

Show that 2ˣ + x³ + x = 10 has exactly one real solution, and find which consecutive integers it lies between.

Check your understanding

Quiz

f is even and g is odd. What is fg?

Which function is even?

For which function is h(−x) = −h(x) for every x?

f(x) = x and g(x) = x are both increasing on ℝ. On which interval is fg = x² decreasing?

Why can y = log x never be an even function?

Which conclusion is correct for y = 2ˣ + x⁵?

For parents and teachers

Big idea: D2.3 asks students to investigate and explain properties of combined functions: odd, even or neither, and increasing or decreasing. The proofs are short and use only the definitions, which is good practice for university mathematics.

How to help: ask "what does f(−x) give?" and have your teen simplify slowly, one sign at a time. For increasing and decreasing, ask for a counterexample whenever a rule seems too good to be true.

Common difficulties:

Thinking odd + odd = even (it is odd; the +1/−1 trick only works for products and quotients).
Concluding "neither" because f(−x) looks different before it is simplified.
Forgetting that the domain must be symmetric.
Using the product rule for increasing functions when the factors are negative.

The Challenge proof (every function is even + odd) is a classic first-year university exercise; working through it slowly is worthwhile.