# Functions: roots, composition and inverses · Grade 10 · 5 units

> Explanation, worked examples and free practice following the Israeli curriculum.

In Functions: roots, composition and inverses (Grade 10 · 5 units) you practise 10 skills: Roots and repeated roots · The degree and the roots · Basic functions · The reciprocal of a function · Composing functions · The inverse function · Roots and fractional powers · Operations on functions · Inequalities by intervals · Limits, intuitively.

A root of a function is a value of x where the function is 0. When a polynomial is written as a product of factors, each factor gives a root. A squared factor gives a repeated root: there the graph touches the axis without crossing it. A polynomial of odd degree always has at least one root, because its two ends have opposite signs.

In a composition you put one function inside another and work from the inside out. The inverse function works the other way: from each value it gives back the x it came from. A function has an inverse only when each value comes from one x alone, so sometimes the domain is cut down, as with a parabola where only x of 0 or more is kept.

To solve an inequality, find the roots and test the sign at one point in each interval between them. A limit describes where a function is heading: a fixed number divided by a number that grows without bound gets closer to 0, and a fixed number divided by a number that gets closer to 0 grows without bound in size.

**Tip:** Before you choose an answer, put one number in and check it: it is the quickest way to catch a sign mistake.

## What you learn

### Roots and repeated roots

- Example: How many roots does the function f(x) = −(x − 6)(x + 5) have?
- Answer: 2
- Solution: Each factor is 0 at one root: x = 6 and x = −5 . The roots are all different, so the function has 2 roots.

### The degree and the roots

- Example: A polynomial function has degree 3. Must it have a root?
- Options: Not necessarily · Yes, it must
- Answer: Yes, it must
- Solution: The degree 3 is odd, so the ends of the graph go opposite ways and the graph crosses the axis. So yes, it must have a root.

### Basic functions

- Example: Is the function f(x) = −|x| even, odd, or neither?
- Options: Odd · Neither even nor odd · Even
- Answer: Even
- Solution: Here |−x| = |x| , so putting in −x gives the same function again. So the function is even.

### The reciprocal of a function

- Example: Here is the function f(x) = x² + 5x − 5 . Find (1 / f)(0) .
- Answer: −1/5
- Solution: First find f(0) = 0² + 5 × 0 − 5 = −5 . So (1 / f)(0) = 1 / (−5) = −1/5 .

### Composing functions

- Example: Here are the functions f(x) = 3x − 4 and g(x) = x² + 3x + 2 . Find f(g(−2)) .
- Answer: −4
- Solution: Inside: g(−2) = (−2)² + 3 × (−2) + 2 = 0 . Then f(0) = 3 × 0 − 4 = −4 .

### The inverse function

- Example: Find the inverse function of f(x) = 2x + 9 .
- Answer: 0.5x − 4.5
- Solution: Write y = 2x + 9 and swap the variables: x = 2y + 9 . Make y the subject: y = (x − 9) / 2 . So f⁻¹(x) = 0.5x − 4.5 .

### Roots and fractional powers

- Example: Work out √((−12)²) .
- Answer: 12
- Solution: First (−12)² = 144 , so √((−12)²) = √144 = 12 . A square root is never negative, so the answer is positive.

### Operations on functions

- Example: Here are the functions f(x) = −2x² + 5 and g(x) = 2x − 4 . Find (f + g)(−2) .
- Answer: −11
- Solution: First f(−2) = −2 × (−2)² + 5 = −3 and g(−2) = 2 × (−2) − 4 = −8 . So (f + g)(−2) = −3 + (−8) = −11 .

### Inequalities by intervals

- Example: Here is the function f(x) = x(x + 5)(x − 6) . What is the sign of f(3) ?
- Options: Positive · Negative
- Answer: Negative
- Solution: Put it in: f(3) = 3(3 + 5)(3 − 6) = −72 . The result is negative.

### Limits, intuitively

- Example: What happens to the function f(x) = −5 / x² as x → ∞ ?
- Options: It grows without bound · It falls without bound · It approaches 1 · It approaches 0
- Answer: It approaches 0
- Solution: The bottom grows without bound in size, and the top is fixed. So it approaches 0.

## Not on the site yet

- Sketching graphs by hand
- Proving that a polynomial of odd degree has a root

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 10 · 5 units topics](https://kixplo.com/il/en/math/grade-10-5u/)