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Functions: domain and range
This test develops understanding of function inputs, outputs, domain and range. Students also find zeros, determine intervals where a function is positive or negative and interpret information given by formulas or graphs.
Questions: 10Estimated time: 18 minutes
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Question 1out of 10
Give the domain of f(x) = 1/(x).
A denominator cannot be zero. For denominator x − a, the excluded value is x = a. Thus x ≠ a, and the domain consists of all real numbers except a.
Question 2out of 10
Given f(x) = 8x − 6, find x when f(x) = −62.
Solve ax + b = y using inverse operations: subtract b from y, then divide by nonzero a. Thus x = (y − b)/a.
Question 3out of 10
Given f(x) = 1x − 6, find f(0).
The notation f(x) represents the function value obtained for a particular value of x.
- Substitute the given number for every x.
- Perform the multiplication first.
- Then add the constant term.
If x or the constant term is negative, use brackets during substitution so that its sign is preserved.
Question 4out of 10
What is the range of f(x) = x²?
The value of f(x) = x² is the square of a real number.
- Remember that the square of a positive number is positive, and the square of a negative number is positive too.
- The smallest possible value occurs when x = 0; it is zero.
- As |x| grows, square values can grow without bound.
Therefore zero must be included in the interval and there is no upper bound. Do not confuse the range with the domain: this asks which y values the function can take.
Question 5out of 10
Find where f(x) = 8x − 48 crosses the x-axis.
A graph crosses the x-axis where its y-coordinate equals zero.
For y = ax + b, substitute y = 0 and solve:
ax + b = 0, so x = −b/a.
The intersection point is written as (x, 0). Check the answer by substituting the calculated x into the original function.
Question 6out of 10
Given f(x) = 8x + 2, find x when f(x) = 10.
Solve ax + b = y using inverse operations: subtract b from y, then divide by nonzero a. Thus x = (y − b)/a.
Question 7out of 10
Which condition is required for a table or mapping to define a function?
The domain is the set of allowed input values x. A function assigns exactly one y to each x in its domain. Different inputs may share an output, but one input cannot have two different outputs.
Question 8out of 10
On which interval is f(x) = (x + 2)(x − 4) negative?
For x₁ < x₂, (x − x₁)(x − x₂) is negative between the roots, where one factor is positive and the other negative. Thus x ∈ (x₁, x₂). Exclude the roots because the function is zero there.
Question 9out of 10
What exactly does the domain of a function mean?
The domain is the set of allowed input values x. A function assigns exactly one y to each x in its domain. Different inputs may share an output, but one input cannot have two different outputs.
Question 10out of 10
Give the domain of f(x) = 1/(x − 1).
A denominator cannot be zero. For denominator x − a, the excluded value is x = a. Thus x ≠ a, and the domain consists of all real numbers except a.
Time left: 00:00:00
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Sample questions
- Given f(x) = ___x + (___), find f(___).
- Given f(x) = ___x + (___), find x when f(x) = ___.
- Give the domain of f(x) = 1/(x − (___)).
- What is the range of f(x) = x²?
- Find where f(x) = ___x + (___) crosses the x-axis.
Functions: domain and range
The domain is the set of permitted input values x, while the range is the set of output values y attained by the function. On a graph, read the domain horizontally and the range vertically.
Finding restrictions
A denominator cannot be zero, and the radicand of an even root must be non-negative. Include an interval endpoint only when the function is defined there. An open or closed point on a graph also indicates whether a boundary value belongs to the set.
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