Functions: roots, composition and inverses · Grade 10 · 5 units
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
For example: How many roots does the function f(x) = −(x − 6)(x + 5) have?
Answer and solution
The answer: 2
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
For example: A polynomial function has degree 3. Must it have a root?
- Not necessarily
- Yes, it must
Answer and solution
The answer: Yes, it must
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
For example: Is the function f(x) = −|x| even, odd, or neither?
- Odd
- Neither even nor odd
- Even
Answer and solution
The answer: Even
Here |−x| = |x| , so putting in −x gives the same function again. So the function is even.
The reciprocal of a function
For example: Here is the function f(x) = x² + 5x − 5 . Find (1 / f)(0) .
Answer and solution
The answer: −1/5
First find f(0) = 0² + 5 × 0 − 5 = −5 . So (1 / f)(0) = 1 / (−5) = −1/5 .
Composing functions
For example: Here are the functions f(x) = 3x − 4 and g(x) = x² + 3x + 2 . Find f(g(−2)) .
Answer and solution
The answer: −4
Inside: g(−2) = (−2)² + 3 × (−2) + 2 = 0 . Then f(0) = 3 × 0 − 4 = −4 .
The inverse function
For example: Find the inverse function of f(x) = 2x + 9 .
Answer and solution
The answer: 0.5x − 4.5
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
For example: Work out √((−12)²) .
Answer and solution
The answer: 12
First (−12)² = 144 , so √((−12)²) = √144 = 12 . A square root is never negative, so the answer is positive.
Operations on functions
For example: Here are the functions f(x) = −2x² + 5 and g(x) = 2x − 4 . Find (f + g)(−2) .
Answer and solution
The answer: −11
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
For example: Here is the function f(x) = x(x + 5)(x − 6) . What is the sign of f(3) ?
- Positive
- Negative
Answer and solution
The answer: Negative
Put it in: f(3) = 3(3 + 5)(3 − 6) = −72 . The result is negative.
Limits, intuitively
For example: What happens to the function f(x) = −5 / x² as x → ∞ ?
- It grows without bound
- It falls without bound
- It approaches 1
- It approaches 0
Answer and solution
The answer: It approaches 0
The bottom grows without bound in size, and the top is fixed. So it approaches 0.
Not on the site yet
Parts of this topic that are not built 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, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).