Precalculus: families of functions · Grade 10 · 4 units
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
For example: Find the domain of the function: f(x) = 1 ÷ (x − 6)
- x > 6
- x ≠ −6
- Every number
- x ≠ 6
Answer and solution
The answer: x ≠ 6
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
For example: Is the function f(x) = x² + 5 even, odd, or neither?
- Odd
- Neither even nor odd
- Even
Answer and solution
The answer: Even
Put in −x for x : f(−x) = (−x)² + 5 = x² + 5 . That is f(x) . So the function is: Even .
Power functions
For example: Which of these functions increases for every value of x ?
- f(x) = x²
- f(x) = x⁴
- f(x) = x³
Answer and solution
The answer: f(x) = x³
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
For 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 and solution
The answer: (5, −1)
The y coordinate goes up by 3, and the x coordinate stays the same: (5, −1)
Reflecting a graph
For 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 and solution
The answer: (5, 4)
The y coordinate changes sign, and the x coordinate stays the same: (5, 4)
Stretching a graph
For example: Here is the function f(x) = x² + 5x − 5 . Write the function g(x) = 2 × f(x) .
Answer and solution
The answer: 2x² + 10x − 10
Multiply every term by 2: g(x) = 2(x² + 5x − 5) = 2x² + 10x − 10
The absolute value of a function
For example: Here is the function f(x) = x² − 6x + 3 . Find |f(1)| .
Answer and solution
The answer: 2
f(1) = 1² − 6 × 1 + 3 = −2 |f(1)| = |−2| = 2
Reading graphs
For example:
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 and solution
The answer: (0, 3)
The graph meets the vertical axis at (0, 3)
Not on the site yet
Parts of this topic that are not built yet: Drawing graphs by hand · Inverse and composite functions.
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 4 units (grades 10 to 12).