Let $f(x)=\frac{\alpha x}{x+1}, x \neq-1$. Then, for what value of $\alpha$ is $f(f(x))=x ?$
[question] Question. Let $f(x)=\frac{\alpha x}{x+1}, x \neq-1$. Then, for what value of $\alpha$ is $f(f(x))=x ?$ (a) $\sqrt{2}$ (b) $-\sqrt{2}$ (c) 1 (d) $-1$ [/question] [solution] Solution: (d) $-1$ $f(f(x))=x$ $\Rightarrow f\left(\frac{\alpha x}{x+1}\right)=x$ $\Rightarrow \frac{\alpha\left(\frac{a x}{x+1}\right)}{\left(\frac{a x}{x+1}\right)+1}=x$ $\Rightarrow \frac{\alpha^{2} x}{\alpha x+x+1}=x$ $\Rightarrow \alpha^{2} x=\alpha x^{2}+x^{2}+x$ $\Rightarrow \alpha^{2} x-\alpha x^{2}-x^{2}-x=...
Let $f(x)=x^{2}$ and $g(x)=2^{x}$. Then, the solution set of the equation $f o g(x)=g o f(x)$ is
[question] Question. Let $f(x)=x^{2}$ and $g(x)=2^{x}$. Then, the solution set of the equation $f o g(x)=g o f(x)$ is [/question] [solution] Solution: Since $(f o g)(x)=(g o f)(x)$, $f(g(x))=g(f(x))$ $\Rightarrow f\left(2^{x}\right)=g\left(x^{2}\right)$ $\Rightarrow\left(2^{x}\right)^{2}=2^{x^{2}}$ $\Rightarrow 2^{2 x}=2^{x^{2}}$ $\Rightarrow x^{2}=2 x$ $\Rightarrow x^{2}-2 x=0$ $\Rightarrow x(x-2)=0$ $\Rightarrow x=0,2$ $\Rightarrow x \in\{0,2\}$ So, the answer is (c). [/solution]...
If $f(x)=\sqrt{x+3}$ and $g(x)=x^{2}+1$ be two real functions,
[question] Question. If $f(x)=\sqrt{x+3}$ and $g(x)=x^{2}+1$ be two real functions, then find fog and gof. [/question] [solution] Solution: $f(x)=\sqrt{x+3}$ For domain, $x+3 \geq 0$ $\Rightarrow x \geq-3$ Domain of $f=[-3, \infty)$ Since $f$ is a square root function, range of $f=[0, \infty)$ $f:[-3, \infty) \rightarrow[0, \infty)$ $g(x)=x^{2}+1$ is a polynomial. $\Rightarrow g: R \rightarrow R$ Computation of fog: Range of $g$ is not a subset of the domain of $f$. and domain $(f o g)=\{x: x \i...
If $f(x)=\sqrt{1-x}$ and $g(x)=\log _{e} x$ are two real functions, then describe functions fog and $g \circ f$.
[question] Question. If $f(x)=\sqrt{1-x}$ and $g(x)=\log _{e} x$ are two real functions, then describe functions fog and $g \circ f$. [/question] [solution] Solution: $f(x)=\sqrt{1-x}$ For domain, $1-\mathrm{x} \geq 0$ $\Rightarrow x \leq 1$ $\Rightarrow$ domain of $f=(-\infty, 1]$ $\Rightarrow f:(-\infty, 1] \rightarrow(0, \infty)$ $g(x)=\log _{e} x$ Clearly, $g:(0, \infty) \rightarrow R$ Computation of $\mathrm{fog}$ : Clearly, the range of $g$ is not a subset of the domain of $f$. So, we need...
Let $f(x)=x^{2}+x+1$ and $g(x)=\sin x$. Show that fog $\neq$ gof.
[question] Question. Let $f(x)=x^{2}+x+1$ and $g(x)=\sin x$. Show that fog $\neq$ gof. [/question] [solution] Solution: $(f o g)(x)=f(g(x))=f(\sin x)=\sin ^{2} x+\sin x+1$ and $(g o f)(x)=g(f(x))=g\left(x^{2}+x+1\right)=\sin \left(x^{2}+x+1\right)$ So, $f o g \neq g o f$. [/solution]...
Let $A=\{-1,0,1\}$ and $f=\left\{\left(x, x^{2}\right): x \in A\right\}$. Show that $f: A \rightarrow A$ is neither one-one nor onto.
[question] Question. Let $A=\{-1,0,1\}$ and $f=\left\{\left(x, x^{2}\right): x \in A\right\}$. Show that $f: A \rightarrow A$ is neither one-one nor onto. [/question] [solution] Solution: $A=\{-1,0,1\}$ and $f=\left\{\left(x, x^{2}\right): x \in A\right\}$ Given, $f(x)=x^{2}$ Given, $f(x)=x^{2}$ Injectivity: $f(1)=1^{2}=1$ and $f(-1)=(-1)^{2}=1$ $\Rightarrow 1$ and $-1$ have the same images. So, $f$ is not one-one. Surjectivity: Co-domain of $f=\{-1,0,1\}$ $f(1)=1^{2}=1$ $f(-1)=(-1)^{2}=1$ and $...
Prove that the function $f: N \rightarrow N$, defined by $f(x)=x^{2}+x+1$, is one-one but not onto.
[question] Question. Prove that the function $f: N \rightarrow N$, defined by $f(x)=x^{2}+x+1$, is one-one but not onto. [solution] Solution: $f: N \rightarrow N$, defined by $f(x)=x^{2}+x+1$ Injectivity: Let x and y be any two elements in the domain (N), such that f(x) = f(y). $\Rightarrow x^{2}+x+1=y^{2}+y+1$ $\Rightarrow\left(x^{2}-y^{2}\right)+(x-y)=0$ $\Rightarrow(x+y)(x-y)+(x-y)=0$ $\Rightarrow(x-y)(x+y+1)=0$ $\Rightarrow x-y=0 \quad[(\mathrm{x}+\mathrm{y}+1)$ cannot be zero because $x$ an...
Let $R=\left\{\left(a, a^{3}\right): a\right.$ is a prime number less than 5$\}$ be a relation.
[question] Question. Let $R=\left\{\left(a, a^{3}\right): a\right.$ is a prime number less than 5$\}$ be a relation. Find the range of $R$. [CBSE 2014] [/question] [solution] Solution: We have, $R=\left\{\left(a, a^{3}\right): a\right.$ is a prime number less than 5$\}$ Or, $R=\{(2,8),(3,27)\}$ So, the range of $R$ is $\{8,27\}$. [/solution]...