Here you will learn what are increasing and decreasing function with examples.<\/p>\n
Let’s begin –<\/p>\n
A function f(x) is said to be a strictly increasing function on (a, b), if<\/p>\n
\n\\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\(f(x_1)\\) < \\(f(x_2)\\) for all \\(x_1\\), \\(x_2\\) \\(\\in\\) (a, b)<\/p>\n<\/blockquote>\n
Thus, f(x) is strictly increasing on (a, b) if the values of f(x) increase with the increase in the values of x.<\/p>\n
Example<\/span><\/strong> : Show that the function f(x) = 2x + 3 is strictly increasing function on R.<\/p>\n
Solution<\/span><\/strong> : Let \\(x_1\\) , \\(x_2\\) \\(\\in\\) R and let \\(x_1\\) < \\(x_2\\). Then,<\/p>\n
\\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\(2x_1\\) < \\(2x_2\\)<\/p>\n
\\(\\implies\\) \\(2x_1\\) + 3 < \\(2x_2\\) + 3<\/p>\n
\\(\\implies\\) \\(f(x_1)\\) < \\(f(x_2)\\)<\/p>\n
Thus, \\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\(f(x_1)\\) < \\(f(x_2)\\) for all \\(x_1\\), \\(x_2\\) \\(\\in\\) R.<\/p>\n
So, f(x) is strictly increasing function on R.<\/p>\n
Example<\/span><\/strong> : Show that the function f(x) = \\(x^2\\) is strictly increasing function on [0, \\(\\infty\\)).<\/p>\n
Solution<\/span><\/strong> : Let \\(x_1\\) , \\(x_2\\) \\(\\in\\) [0, \\(\\infty\\)) and let \\(x_1\\) < \\(x_2\\). Then,<\/p>\n
\\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\((x_1)^2\\) < \\(x_1x_2\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[Multiplying both sides by \\(x_1\\)]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0……(i)<\/p>\n
again, \\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\(x_1x_2\\) < \\((x_2)^2\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[Multiplying both sides by \\(x_2\\)]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0……(ii)<\/p>\n
from (i) and (ii), we get<\/p>\n
\\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\((x_1)^2\\) < \\((x_2)^2\\) \\(\\implies\\) \\(f(x_1)\\) < \\(f(x_2)\\)<\/p>\n
Thus, \\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\(f(x_1)\\) < \\(f(x_2)\\) for all \\(x_1\\), \\(x_2\\) \\(\\in\\) [0, \\(\\infty\\)).<\/p>\n
So, f(x) is strictly increasing function on [0, \\(\\infty\\)).<\/p>\n
Strictly Decreasing Function<\/strong><\/h4>\n
A function f(x) is said to be a strictly decreasing function on (a, b), if<\/p>\n
\n\\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\(f(x_1)\\) > \\(f(x_2)\\) for all \\(x_1\\), \\(x_2\\) \\(\\in\\) (a, b)<\/p>\n<\/blockquote>\n
Thus, f(x) is strictly decreasing on (a, b) if the values of f(x) decrease with the increase in the values of x.<\/p>\n
Example<\/span><\/strong> : Show that the function f(x) = -3x + 12 is strictly decreasing function on R.<\/p>\n
Solution<\/span><\/strong> : Let \\(x_1\\) , \\(x_2\\) \\(\\in\\) R and let \\(x_1\\) < \\(x_2\\). Then,<\/p>\n
\\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\(-3x_1\\) < \\(-3x_2\\)<\/p>\n
\\(\\implies\\) \\(-3x_1\\) + 12 < \\(-3x_2\\) + 12<\/p>\n
\\(\\implies\\) \\(f(x_1)\\) > \\(f(x_2)\\)<\/p>\n
Thus, \\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\(f(x_1)\\) > \\(f(x_2)\\) for all \\(x_1\\), \\(x_2\\) \\(\\in\\) R.<\/p>\n
So, f(x) is strictly decreasing function on R.<\/p>\n
Example<\/span><\/strong> : Show that the function f(x) = \\(a^x\\), 0 < a < 1 is strictly decreasing function on R.<\/p>\n
Solution<\/span><\/strong> : Let \\(x_1\\) , \\(x_2\\) \\(\\in\\) R and let \\(x_1\\) < \\(x_2\\). Then,<\/p>\n
\\(x_1\\) < \\(x_2\\)<\/p>\n
\\(\\implies\\) \\(a^{x_1}\\) < \\(a^{x_2}\\) \\(\\implies\\) \\(f(x_1)\\) > \\(f(x_2)\\)<\/p>\n
Thus, \\(x_1\\) < \\(x_2\\) \\(\\implies\\) \\(f(x_1)\\) > \\(f(x_2)\\) for all \\(x_1\\), \\(x_2\\) \\(\\in\\) R.<\/p>\n
So, f(x) is strictly decreasing function on R.<\/p>\n
\nRelated Questions<\/h3>\n
Find the interval in which f(x) = \\(-x^2 \u2013 2x + 15\\) is increasing or decreasing.<\/a><\/p>\n