Question

Which statements are true of functions? Check all that apply.

All functions have a dependent variable.
All functions have an independent variable.
The range of a function includes its domain.
A vertical line is an example of a functional relationship.
A horizontal line is an example of a functional relationship.
Each output value of a function can correspond to only one input value.

Answer 1

True:
- All functions have a dependent variable.
- A function assigns outputs (dependent values) to inputs, so there is always a dependent variable.
- All functions have an independent variable.
- A function has inputs (independent variable(s)) even if the output is constant.
- A horizontal line is an example of a functional relationship.
- y = c (a horizontal line) gives exactly one y for each x (a constant function).

False:
- The range of a function includes its domain.
- The range is the set of outputs; the domain is the set of inputs. They are generally different (e.g., f(x)=x^2 with domain R has range [0,∞), which does not include most domain values).
- A vertical line is an example of a functional relationship.
- A vertical line x = c fails the vertical-line test (one x gives many y’s), so it is not the graph of a function y = f(x).
- Each output value of a function can correspond to only one input value.
- This would require the function to be one-to-one. In general outputs can come from multiple inputs (e.g., f(x)=x^2 gives f(2)=f(−2)=4).