Mathematical language and symbols

What the board asks

Expect items on the characteristics of mathematical language, translation, and evaluating functions.

• Characteristics: precise, concise, powerful.
• Statements: a statement has a definite truth value. An open sentence such as x + 1 = 5 does not until x is given.
• Inverse: swap x and y, then solve for y.
• Range of a quadratic: find its lowest or highest value first.

Solved examples

Follow each one line by line. The steps stay the same when the numbers change, so cover the steps and try it yourself.

Example 1: Finding an inverse

Question: What is the inverse of f(x) = 3x − 5?

1. Write it with y. y = 3x − 5.
2. Swap x and y. x = 3y − 5.
3. Solve for y. x + 5 = 3y, so y = (x + 5) ÷ 3.
4. Check with a number. f(2) = 1, and (1 + 5) ÷ 3 = 2, which returns the input.

Answer: The inverse is (x + 5) ÷ 3.

Example 2: Range of a quadratic

Question: What is the range of f(x) = x² + 6?

1. Find the lowest value of the square. x² is never negative, so its lowest value is 0.
2. Add the rest of the rule. The lowest value of x² + 6 is 0 + 6 = 6.
3. State the range. The square can grow without limit, so every value from 6 upward is reached.

Answer: The range is y ≥ 6.

Example 3: A made-up operation

Question: If a # b = 2a − b + ab, what is 3 # 4?

1. Match the letters to the numbers. a = 3 and b = 4.
2. Substitute exactly as defined. 2(3) − 4 + (3)(4).
3. Compute. 6 − 4 + 12 = 14.

Answer: 3 # 4 = 14.