ACT Math · Lesson 5 of 5
Functions, statistics, and probability
Read a function as a rule with a domain, and use the correct denominator in a probability or a statistic.
Functions and transformations
A function assigns each input in its domain to one output. Evaluating asks for the output. Solving asks which inputs produce a given output. A vertical shift f(x) + 3 adds 3 to every output. A horizontal shift f(x − 3) replaces the input with a number 3 less, which moves the graph right.
A logarithm is the inverse of an exponential. log₁₀(1000) = 3 because 10³ = 1000. The argument of a logarithm must be positive.
Statistics and probability
The mean is pulled by extreme values. The median is the middle of an ordered list and may stay put when one extreme changes. Probability of an event is favorable outcomes over possible outcomes in the situation that is actually being described.
“Given that” restricts the possible outcomes. If 9 of 24 bus riders are in a club, and you already know the student rides the bus, the probability is 9/24, not 9 divided by the whole school.
Worked example
A bag has 5 red and 7 blue marbles. One marble is drawn and not replaced; it is red. What is the probability the next one is also red?
- After the first red marble, 4 red marbles remain and 11 marbles remain in total.
- The new probability is 4/11.
- The original probability of red was 5/12, which no longer describes the bag.
Why this works. A condition that removes an outcome changes both the numerator and the denominator when that outcome was one of the counted items.
Check your understanding
- Describe a transformation as a change to the input or to the output.
- Choose mean or median according to whether every value, or the middle, matters.
- Rewrite the sample space after a “given that” condition.