الأسبوع 4 · إشارة الدالة الثابتة والخطية والتربيعية، ومتباينات الدرجة الثانية في مجهول واحد · 3 دروس
1
The sign of a constant and of a linear function: where f(x) is positive, negative or zero
Mathematics Sec 1 Term 1, Unit 1, Lessons 4–5
GoalFind the sign of a constant function and of a linear function over the real numbers.
New words
sign of a function · the intervals where f(x) is positive, negative or zerozero of a function · a value of x where f(x) = 0open interval · ]a, b[ : the numbers between a and b, without a and b
Explanation
To study the sign of a function means to say for which values of x the value f(x) is positive, negative or zero. A constant function f(x) = c has the sign of c for every x. A linear function f(x) = bx + c (b ≠ 0) has one zero, at x = −c/b. To the right of the zero it has the sign of b, and to the left the opposite sign. Take f(x) = 6 − 3x: the zero is x = 2, and b = −3 is negative, so f is negative for x > 2 and positive for x < 2. The solution sets are written as intervals: ]2, ∞[ means all numbers greater than 2, ]−∞, 2[ all numbers less than 2. A quick check: substitute one number from each side. f(0) = 6 > 0 and f(3) = −3 < 0 ✓. When two conditions must hold together, take the numbers that satisfy both, the intersection of the two sets.
x
x < 2
x = 2
x > 2
6 − 3x
+
0
−
x − 2
−
0
+
Worked examples
Sign of f(x) = 6 − 3x. Zero: 6 − 3x = 0 gives x = 2. The coefficient of x is −3 < 0. So f(x) > 0 on ]−∞, 2[, f(2) = 0 and f(x) < 0 on ]2, ∞[.
Sign of f(x) = 2x + 5. Zero: x = −5/2. The coefficient 2 is positive, so f(x) > 0 for x > −5/2 and f(x) < 0 for x < −5/2.
For which x are f(x) = x + 3 and g(x) = 5 − x both positive? f(x) > 0 for x > −3. g(x) > 0 for x < 5. Both at once: the intersection ]−3, 5[. (Joining the two sets would give all real numbers, which is wrong.)
Class activity
Sign walk. Draw a number line. For f(x) = 3x − 6 mark the zero, then test one number on the left and one on the right and write + or − above each side.
Practice
1
Find the sign of f(x) = 3x − 6.
Zero at x = 2; positive for x > 2, negative for x < 2.
2
Find the sign of f(x) = −4x − 8.
Zero at x = −2; the coefficient −4 is negative, so f is negative for x > −2 and positive for x < −2.
3
What is the sign of f(x) = 7 and of g(x) = −3?
f is positive for every x; g is negative for every x.
4
For which x is 5 − 2x negative?
5 − 2x = 0 at x = 5/2; the coefficient −2 is negative, so it is negative for x > 5/2.
5
For which x are x − 1 and 9 − x² both positive?
x − 1 > 0 on ]1, ∞[. 9 − x² > 0 on ]−3, 3[. Both: ]1, 3[.
