What Is a Function? — Challenge
A function is a rule with no ambiguity: one input, exactly one output. That is the whole definition.
What these problems cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Function — A rule pairing each input with exactly one output.
- Slope — How much y changes for each 1 that x increases.
- Domain — The set of allowed inputs.
- Range — The set of outputs that actually occur.
- Parabola — The U-shaped graph of a quadratic function.
Problem 1
Which of these sets of ordered pairs is a function?
- {(1,2),(2,3),(3,4)}
- {(1,2),(1,3)}
- {(2,1),(2,5)}
- {(0,1),(0,2)}
- {(4,1),(4,2)}
Worked answer
{(1,2),(2,3),(3,4)}
A function gives each input exactly one output.
In the first set the inputs 1, 2 and 3 each appear once, so each has a single output.
Every other set repeats an input with two different outputs, which no function may do.
Problem 2
Which of these equations does not define y as a function of x?
- y=x2
- y=2x
- x=y2
- y=x+1
- y=|x|
Worked answer
x=y2
A function of x gives one y for each x.
In x = y2, taking x=4 gives y=2 or y=-2, two outputs for one input.
Every other equation produces a single y once x is chosen, even y=x2, where different inputs may share an output but no input has two.
Problem 3
What is the domain of f(x)=1x-3?
- All real numbers
- All real numbers except 3
- All real numbers except 0
- Only positive numbers
- Only x>3
Worked answer
All real numbers except 3
The domain is every input the rule can accept.
Division by zero is undefined, so the denominator x-3 must not be zero, which rules out x=3.
Every other real number works, so the domain is all real numbers except 3.
Problem 4
If f(x)=x+5, what is f(2)?
- 3
- 7
- 10
- 25
- 2
Worked answer
7
Substitute 2 wherever x appears in the rule f(x)=x+5.
f(2) = 2+5 = 7.
The notation f(2) names the output for the input 2; it does not mean f multiplied by 2.
Problem 5
What is the set of all outputs of a function called?
- The domain
- The range
- The slope
- The intercept
- The variable
Worked answer
The range
The inputs a function accepts form its domain.
The values it produces form its range.
So the set of outputs is the range.
Problem 6
If f(x)=2x, what is f(0)?
- 0
- 1
- 2
- -2
- Undefined
Worked answer
0
Substitute 0: f(0) = 2· 0.
Anything multiplied by zero is zero, so f(0) = 0.
Zero is a perfectly ordinary input here; nothing is undefined.
Problem 7
A vertical line crosses a graph at two points. Is the graph a function?
- Yes
- No
- Only if the line is the y-axis
- It cannot be decided
- Only for straight lines
Worked answer
No
Every point on a single vertical line shares the same x value.
Two crossing points therefore mean one input has two different outputs.
That is exactly what a function may not do, so the graph is not a function. This is the vertical line test.
Problem 8
What is the set of all inputs of a function called?
- The domain
- The range
- The output
- The image
- The codomain
Worked answer
The domain
The values a function is allowed to take in are its domain.
The values it gives out are its range.
So the set of inputs is the domain.