SAT-47: Functions — Domain, Range, and Evaluation

SAT-47: Functions — Domain, Range, and Evaluation

Description: A function is a rule that takes an input and gives exactly one output. This lesson covers function notation f(x), how to evaluate it, and the meaning of domain and range.

Function notation

f(x) means "the output of f for input x." To evaluate, just substitute the number for x.

(Oʻzbekcha: funksiya — har bir kirish uchun aniq bitta chiqish beruvchi qoida.)

Domain and range

• Domain = all allowed inputs (x-values).
• Range = all possible outputs (y-values).

Two common domain limits: you can't divide by zero, and you can't take the square root of a negative number. (Oʻzbekcha: aniqlanish sohasi — ruxsat etilgan x lar; qiymatlar sohasi — chiqadigan y lar.)

Worked Example (evaluate)

If f(x) = 2x² − 3, find f(4) and f(−1).

• f(4) = 2(4)² − 3 = 32 − 3 = 29.
• f(−1) = 2(−1)² − 3 = 2 − 3 = −1.

Worked Example (domain)

What is the domain of f(x) = 1/(x − 5)?

• The denominator can't be 0, so x − 5 ≠ 0 → x ≠ 5.
• Domain: all real numbers except 5.

(Oʻzbekcha: x = 5 da maxraj 0 boʻladi, shuning uchun u sohaga kirmaydi.)

Practice

If g(x) = x² + 1, what is the smallest value in its range?

Key words — Kalit soʻzlar

• Function — funksiya
• Function notation f(x) — funksiya belgisi f(x)
• Input — kirish (argument)
• Output — chiqish (qiymat)
• Evaluate — qiymatini hisoblash
• Domain — aniqlanish sohasi
• Range — qiymatlar sohasi
• Substitute — oʻrniga qoʻyish
• Real numbers — haqiqiy sonlar

Summary

• f(x) is a rule with one output per input; evaluate by substituting.
• Domain = allowed inputs; range = possible outputs.
• Exclude inputs that cause division by zero or a square root of a negative.