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Composition of function, Condition for Composite Function, Property of Composite Function - (Concept)
Let $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ and $\mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}$ be two functions. Then the composition of f and g is denoted by gof and defined as the function gof : $\mathrm{A} \rightarrow \mathrm{C}$ given by $\operatorname{gof}(x)=g(f(x))$
Properties of composition:
In general fog $\neq$ gof (Not commutative)
$\mathrm{fo}(\mathrm{goh})=(\mathrm{fog})$ oh $\quad$ (Associative $)$
If $f$ and $g$ are bijections then fog and gof are also bijections
The composition of any function with the identity function is the function itself. If $f: A \rightarrow B$, then $f o I_A=I_B o f=f$
| Exam | Chapter |
| JEE MAIN | Sets, Relations and Functions |
For then
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If h(x)=3x-5; . Find goh(x)
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$\begin{aligned} & \text { For } x \in \mathbb{R}-\{0,1\}, \text { let } f_1(x)=\frac{1}{x}, f_2(x)=1-x \text { and } \\ & f_3(x)=\frac{1}{1-x} \text { be three given functions. If a function } J(x) \\ & \text { satisfies }\left(f_2 \circ J \circ f_1\right)(x)=f_3(x), \text { then } J(x) \text { is equal to: }\end{aligned}$
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Let For any
, define
. If
so, then which one of the following statements is not true?
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For let
and
. If
then
is equal to :
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If and
are both onto functions then
is
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If then which of the following
satisfies
?
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If then find
such that
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If and
, then
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What is the inverse of ?
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What is inverse of ?
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The inverse of the function
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If f(x) and g(x) are two functions. are domains and range of f(x) and g(x) respectively. Then which of the following is true if fog exists?
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If gof and fog both exist, then which of these is not true?
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If
. Then fog=
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In which of the cases of a pair of functions, the composition of the function is commutative?
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If f(x) = 2x and g(x) = 4x, then
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If
, then evaluate gof(x) + fog(x)
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If f(x) = 2x, g(x) = , then
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If and
. Also
. Then find
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If , then find (gof)oh(x)
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If and
then
is equal to:
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If f is a one one function but g is a many one function where and
then
is a
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Let be a function which satisfies
then the value of n for which
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If and
. Then the value of
is?
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Q: For a suitably chosen real constant a, let a function, $f: R-\{-a\} \rightarrow R_{\text {be }}$ defined by $f(x)=\frac{a-x}{a+x}$. Further suppose that for any real number $x \neq-a$ and $f(x) \neq-a,(f \circ f)(x)=x$. Then $f\left(-\frac{1}{2}\right)$ is equal to:
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Let $f: R \rightarrow R {\text {be defined as }} f(x)=2 x-1$ and $g: R-\{1\} \rightarrow R_{\text {be }}$ defined as $g(x)=\frac{x-\frac{1}{2}}{x-1}$.Then the composition function $f(g(x))$ is:
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Let and
. If
, then the domain of the function fog is:
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For, let
and
then the value of
is approximately equal to:
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Let be two functions defined by
and
. Then, for which of the following range of
, the inequality
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Let be a function defined by
. If the function
, then the greatest integer less than or equal to
is______________.
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Let is equal to.
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Let be defined as
and
be defined as
Then the function
is.
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Let . Define
it as
Let be a function such that
Then
is equal to ___________.
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Let be two real polynomials of degree
respectively. If
, then the value of
is:
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Let be a quadratic polynomial with leading coefficient 1 such that
. If the equations
have a common real root, then
is equal to____________
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Let $f: \mathbf{R}-\left\{\frac{\alpha}{6}\right\} \rightarrow \mathbf{R}$ be defined by $f(x)=\frac{5 x+3}{6 x-\alpha}$.
Then the value of for which $(fof) (x)=x$, for all $x \in \mathbf{R}-\left\{\frac{\alpha}{6}\right\}$, is?
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Let $f(x)=2^{10} \cdot x+1 \text { and } g(x)=3^{10} \cdot x-1 \text {. If }(f \circ g)(x)=x$, then x is equal to:
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If $f(x)=c$ where $c \epsilon R$ and $g(x)=e^x$ then $f o g(x)=$
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Let N be the set of natural numbers and two functions and
be defined as
such that
and
. Then
is:
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Consider the following statements :
$S_1: x=\sqrt{\log _{11} 7}$ and $y=\sqrt{\log _7 11}$, then $e^{y \ln 7-x \ln 7}$ is equal to 1.
$S_2: \log _x 3>\log _x 2$ is true for all value of $x \epsilon(0,1) \cup(1, \infty)$
$S_3:|x-2|=[-\pi]$, then $x$ is $6,-2$
$S_4: \log _{25}\left(2+\tan ^2 \theta\right)=0.5$, then $\theta$ maybe $\frac{4 \pi}{3}$ or $\frac{2 \pi}{3}$
State, in order, whether $S_1, S_2, S_3, S_4$ are true or false
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For some , let
and
.If
is equal to
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For , two real-valued functions
are such that,
and fog
. Then
is equal to:
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If $\mathrm{f}(\mathrm{x})=\frac{4 \mathrm{x}+3}{6 \mathrm{x}-4}, \mathrm{x} \neq \frac{2}{3}$ and (fof) (x) $\mathrm{g}(\mathrm{x})$, where $\mathrm{g}: \mathbb{R}-\left\{\frac{2}{3}\right\} \rightarrow \mathbb{R}-\left\{\frac{2}{3}\right\}$, then (gogog)(4) is equal to
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Let $f(x)=\left\{\begin{array}{ll}x-1, & x \text { is even, } \\ 2 x, & x \text { is odd, }\end{array} x \in\right.$. If for some $a \in N, f(f(f(a)))=21$, then $\lim _{x \rightarrow a^{-}}\left\{\frac{|x|^3}{a}-\left[\frac{x}{a}\right]\right\}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to :
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| D. |
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Let , then the domain of the function
is
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Consider the function $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $\mathrm{f}, \underbrace{(f \text { o f o f o...o })}_{10 \text { times }}(\mathrm{x})=\frac{2^{10} \mathrm{x}}{\sqrt{1+9 \alpha \mathrm{x}^2}}$, then the value of $\sqrt{3 \alpha+1}$ is equal to .....
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$f(x)=x^2$ and $g(x)=\sin x$, find $\operatorname{gof}(x)$
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For $\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$, find the calculation for which $\mathrm{fog}(\mathrm{x})$ can be evaluated.
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Identify the correct statement:
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If $(1,2) \in S o R, \mathrm{~S} \& \mathrm{R}$ are 2 relations from B to C and A to B respectively, then there exists $b \in B$, such that:
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If $f(x)=\frac{1}{\sqrt{x}}, g(x)=x-2$, then the domain of $f \circ g(x)$ is
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If $f=\{(1,2),(3,4),(5,6)\}$ and $g=\{(2,4),(3,5),(4,1)\}$, then the value of $f \circ g(3)+g \circ f(1)$ is
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| D. |
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If P is the number of prime numbers less than or equal to 52, and q is the numbers composite numbers less than or equal to 52, then p + q equals
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| D. |
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Let $f, \mathrm{~g}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as : $f(\mathrm{x})=|\mathrm{x}-1|$ and $g(x)=\left\{\begin{array}{cc}\mathrm{e}^{\mathrm{x}}, & \mathrm{x} \geq 0 \\ \mathrm{x}+1, & \mathrm{x} \leq 0\end{array}\right.$. Then the function $f(\mathrm{~g}(\mathrm{x}))$ is
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If f and g are two functions from R to R defined as $f(x)=|x|+x$ and $g(x)=|x|-x$, then fog $(\mathrm{x})$ for $\mathrm{x}<0$ is
| A. |
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| C. |
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| D. |
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Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be the bijective functions. Then $(g \circ f)^{-1}$ is
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|
| C. |
|
| D. |
|
Let $f: N \rightarrow R$ be the function defined by $f(x)=(2 x-1) / 2$ and $g: Q \rightarrow R$ be another function defined by $g(x)=x+2$ then, $\operatorname{gof}(3 / 2)$ is
| A. |
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| B. |
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| C. |
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| D. |
|
Let $f: R \rightarrow R$ be the functions defined by $f(x)=x^3+5$. Then $f^{-1}(x)$ is
| A. |
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|
| C. |
|
| D. |
|
Let $f: R \rightarrow R$ be defined by $f(x)=1 / x \forall x \in R$ then f is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $f: R \rightarrow R$ be defined by $f(x)=3 x^2-5$ and $g: R \rightarrow R$ by $g(x)=x /\left(x^2+1\right)$, then gof is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For identity function , $f(x)=x^3-3$ can be used to express following as:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $f(x)=1-x$, then $f(f(f(x)))$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $f(x)=\log \left(x^2+1\right)$ and $g(x)=e^x$, then $f o g(x)$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $f(x)=\cos 5 x+A \cos 4 x+B \cos 3 x+C \cos 2 x+D \cos x+E$, and
$
T=f(0)-f\left(\frac{\pi}{5}\right)+f\left(\frac{2 \pi}{5}\right)-f\left(\frac{3 \pi}{5}\right)+\ldots . .+f\left(\frac{8 \pi}{5}\right)-f\left(\frac{9 \pi}{5}\right)
$
Then $T$
| A. |
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| B. |
|
| C. |
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| D. |
|
If $f(1)=2, f(2)=3, f(3)=5, g(2)=3, \quad g(3)=10, g(5)=51$, then the value gof $(1)+g \circ f(3)$ is
| A. |
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| B. |
|
| C. |
|
| D. |
|
Definition 8 Let f : A B and g : B
C be two functions. Then the composition of
f and g, denoted by gof, is defined as the function gof : A C given by
gof (x) = g(f (x)), x
A.
