# Precalculus: families of functions · Grade 10 · 4 units

> Precalculus: families of functions, Grade 10 · 4 units: The domain and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Precalculus: families of functions (Grade 10 · 4 units) you practise 8 skills: The domain · Even and odd functions · Power functions · Translating a graph · Reflecting a graph · Stretching a graph · The absolute value of a function · Reading graphs.

The domain of a function is every value you are allowed to put in. Under a square root the expression is not negative, and in a fraction the denominator is not zero. An even function stays the same when x is replaced by its opposite, and its graph is symmetric in the y axis. An odd function changes sign, and its graph is symmetric about the origin.

Adding a number to a function moves its graph up or down, and adding a number to x inside the function moves it left or right, in the direction opposite to the sign. Multiplying the function by minus one reflects the graph in the x axis, and replacing x by its opposite reflects it in the y axis. Multiplying the function by a number greater than one stretches the graph away from the x axis.

The absolute value of a function keeps the parts above the x axis and reflects upwards the parts below it. Where a question has a picture, it shows both graphs: the original dashed and the new one solid.

**Tip:** Check one point: take a point on the original graph and see where it goes. That is enough to know which way the graph moved.

## What you learn

### The domain

- Example: Find the domain of the function: f(x) = 1 ÷ (x − 6)
- Options: x > 6 · x ≠ −6 · Every number · x ≠ 6
- Answer: x ≠ 6
- Solution: You cannot divide by 0, so the denominator is not 0: x − 6 ≠ 0 x ≠ 6 So the domain is every number except that one.

### Even and odd functions

- Example: Is the function f(x) = x² + 5 even, odd, or neither?
- Options: Odd · Neither even nor odd · Even
- Answer: Even
- Solution: Put in −x for x : f(−x) = (−x)² + 5 = x² + 5 . That is f(x) . So the function is: Even .

### Power functions

- Example: Which of these functions increases for every value of x ?
- Options: f(x) = x² · f(x) = x⁴ · f(x) = x³
- Answer: f(x) = x³
- Solution: With an even power the graph goes down until x = 0 and then goes up, or the other way round, so it does not fit. With the third power the graph keeps going the same way all along: f(x) = x³

### Translating a graph

- Example: The point (5, −4) is on the graph of y = f(x) . Which point matches it on the graph of y = f(x) + 3 ?
- Answer: (5, −1)
- Solution: The y coordinate goes up by 3, and the x coordinate stays the same: (5, −1)

### Reflecting a graph

- Example: The point (5, −4) is on the graph of y = f(x) . Which point matches it on the graph of y = −f(x) ?
- Answer: (5, 4)
- Solution: The y coordinate changes sign, and the x coordinate stays the same: (5, 4)

### Stretching a graph

- Example: Here is the function f(x) = x² + 5x − 5 . Write the function g(x) = 2 × f(x) .
- Answer: 2x² + 10x − 10
- Solution: Multiply every term by 2: g(x) = 2(x² + 5x − 5) = 2x² + 10x − 10

### The absolute value of a function

- Example: Here is the function f(x) = x² − 6x + 3 . Find |f(1)| .
- Answer: 2
- Solution: f(1) = 1² − 6 × 1 + 3 = −2 |f(1)| = |−2| = 2

### Reading graphs

- Example: (A coordinate grid with the graph of a function: a broken line) The picture shows the graph of the function y = f(x) . The graph carries on in the same direction beyond the picture. At which point does the graph cross the vertical axis?
- Answer: (0, 3)
- Solution: The graph meets the vertical axis at (0, 3)

## Not on the site yet

- Drawing graphs by hand
- Inverse and composite functions

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 4 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 10 · 4 units topics](https://kixplo.com/il/en/math/grade-10-4u/)