Functions are relations in which each element of the domain is paired with exactly one element of the range.
Examples of functions:
f(x) = x2 + x + 1
The domain of f is all real numbers.
g(x) = 1/(x - 2)
The domain of g is all real numbers except x = 2.
h(x) = √x
The domain of h is all real numbers greater than or equal to 0.
Example of SAT question with functions: f(x) = 1/(x + 1) and g(x) = x + 1, x≠-1. What are the value of x for which f(x) = g(x)?
(a) x1 = 0 and x2 = 2
(b) x1 = 0 and x2 = 1
(c) x1 = 2 and x2 = -2
(d) x1 = 0 and x2 = -2
(e) x1 = 1 and x2 = -2
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Answer: f(x) = g(x), x + 1 = 1/(x + 1);
(x + 1) · (x + 1) = 1;
x2 + 2 · x = 0
x · (x + 2) = 0 so x1 = 0 and x2 = -2. (d) is the correct answer.
Some of the algebra questions require factorization of polynomials. Let’s see some examples:
(x - 4)2 = (x - 2)·(x + 2)
x2 + 3·x = x·(x + 3)
a·c·x2 + (a·d + b·c)·x + b·d = (a·x + b)·(c·x + d)
The absolute value of m is defined as the distance from m to zero on the number line.
If m is positive or equal to 0, |m| = m
If m is negative, |m| = -m
Example of SAT question with absolute value:Which value of x satisfies the inequality | 2x | < x + 1 ?
(a) -1/2
(b) 1/2
(c) 1
(d) -1
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Answer: We can write the inequality as -(x + 1) < 2·x < x + 1
The left side: -x - 1 < 2·x; 3·x + 1 >0 so x > -1/3
The right side: 2·x < x + 1; x < 1
The only x of the answers that satisfies the inequality is x = 1/2
You will need to apply the following rules for exponents:
ab·ac = ab + c
ab/ac = ab - c for a ≠ 0
(a·b)c = ac·bc
(a/b)c = ac / bc for b ≠ 0
(ab)c = ab·c
Example of SAT question with exponents: What is the value of (3x + 1 - 3x) / (3x - 3x - 1)?
(a) 3x
(b) 3x + 1
(c) 3x - 1
(d) 3
(e) 6
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Answer: The numerator of the fraction is: 3x + 1 - 3x = 3x·(3 - 1) = 2 · 3x
The denominator of the fraction is: 3x - 3x - 1 = 3x - 1·(3 - 1) = 2 · 3x - 1
We can write the fraction as (2 · 3x) / (2 · 3x - 1) = 3x / 3x - 1 = 3 · 3x - 1 / 3x - 1 = 3
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