(A) Given that $f'(x) > 0$,the function $f$ is strictly increasing.Given that $g'(x) < 0$,the function $g$ is strictly decreasing.Since $g$ is strictl

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(A) Given that $f'(x) > 0$,the function $f$ is strictly increasing.Given that $g'(x) Since $g$ is strictly decreasing,for $x g(x+1)$.Since $f$ is strictly increasing,applying $f$ to both sides of the inequality $g(x) > g(x+1)$ preserves the inequality sign.Therefore,$f(g(x)) > f(g(x+1))$.Thus,option $A$ is correct.

Class 12 Mathematics Relations and Functions Mix Examples of Relation and Function

Let $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$. Then which of the following is true?

A
$f\{g(x)\} > f\{g(x+1)\}$
B
$f\{g(x-1)\} < f\{g(x+1)\}$
C
$g\{f(x-1)\} < g\{f(x+1)\}$
D
$g\{f(x)\} < g\{f(x-1)\}$

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Relations and Functions

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Mix Examples of Relation and Function

Consider the following statements:
$(i)$ $A$ relation is a special case of a function.
$(ii)$ $A$ function is a special case of a relation.
$(iii)$ Both relation and function are the same.

EasyAP EAMCET 2020

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Let $f(x) = \sqrt{x}$ and $g(x) = x$ be two functions defined over the set of non-negative real numbers. Find $(f+g)(x)$,$(f-g)(x)$,$(fg)(x)$,and $(\frac{f}{g})(x)$.

Medium

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Let $f(x) = \frac{x^2 - 4}{x^2 + 4}$ for $|x| > 2$. Then the function $f: (- \infty, -2] \cup [2, \infty) \to (-1, 1)$ is

Easy

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Let $f(x) = a^x$ $(a > 0)$ be written as $f(x) = f_1(x) + f_2(x)$,where $f_1(x)$ is an even function and $f_2(x)$ is an odd function. Then $f_1(x + y) + f_1(x - y)$ equals

DifficultJEE MAIN 2019

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The number of roots of the equation $\sqrt{2}+e^{\cosh^{-1} x}-e^{\sinh^{-1} x}=0$ is

MediumTS EAMCET 2019

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Let $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$. Then which of the following is true?

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Consider the following statements: $(i)$ $A$ relation is a special case of a function. $(ii)$ $A$ function is a special case of a relation. $(iii)$ Both relation and function are the same.

Let $f(x) = \sqrt{x}$ and $g(x) = x$ be two functions defined over the set of non-negative real numbers. Find $(f+g)(x)$,$(f-g)(x)$,$(fg)(x)$,and $(\frac{f}{g})(x)$.

Let $f(x) = \frac{x^2 - 4}{x^2 + 4}$ for $|x| > 2$. Then the function $f: (- \infty, -2] \cup [2, \infty) \to (-1, 1)$ is

Let $f(x) = a^x$ $(a > 0)$ be written as $f(x) = f_1(x) + f_2(x)$,where $f_1(x)$ is an even function and $f_2(x)$ is an odd function. Then $f_1(x + y) + f_1(x - y)$ equals

The number of roots of the equation $\sqrt{2}+e^{\cosh^{-1} x}-e^{\sinh^{-1} x}=0$ is

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Consider the following statements:
$(i)$ $A$ relation is a special case of a function.
$(ii)$ $A$ function is a special case of a relation.
$(iii)$ Both relation and function are the same.