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If \({\log _2}x\) is a logarithmic function then it is equal to,
If \(f = {x^5} – 3{x^4} + 2{x^3} + 9{x^2} – 5x + 9\), then find out which type of function it is?
or
Hence this function is neither odd nor even.
If \(f\left( x \right) = {x^4} – 3{x^2}\), then f(x) is
is an even function.
Let \(f\left( x \right) = {x^n},g\left( x \right) = \sin x\), then\(gof\left( x \right)\) is equal to
If \(f\left( x \right) = \cos ecx,g\left( x \right) = \cos e{c^{ – 1}}x\), then gof(x) is equal to
If \(9 \equiv 2\left( {\bmod 7} \right)\)and\(5 \equivย – 2\left( {\bmod 7} \right)\), then
i.e.,
If \(f\left( x \right) = \frac{1}{{{x^n}}},n \in N\), then domf is equal to
Hence , since is defined.
Domain of is
Range of the function \(f\left( x \right) = \frac{{{x^2} + x + 2}}{{{x^2} + x + 1}},x \in R\) is
Let
i.e.,
Since is real,
Range of is
Let \(f\left( \thetaย \right) = \sin \theta \left( {\sin \thetaย + \sin 3\theta } \right)\), then\(f\left( \thetaย \right)\)
This is true for all real .
\(f\left( x \right) = \sin \log \left( {\frac{{\sqrt {4 – {x^2}} }}{{1 – x}}} \right)\), the Df is
is defined for
Where and
and
and
domain of
If \(f\left( x \right) = \sqrt {{{\sin }^{ – 1}}\left( {2x} \right) + \frac{\pi }{6}} \), then domain of f is
If \({2^x} + {2^y} = 2\) is a function, then domain of function f(x) is
But,
Hence
Taking on both sides with base , we have
The domain of definition of the function\(y = \frac{1}{{{{\log }_{10}}\left( {1 – x} \right)}} + \sqrt {x + 2} \)is
For domain of
and
and
excluding
Let \(E = \left\{ {1,2,3,4} \right\}\) and\(F = \left\{ {1,2} \right\}\), then the number of onto function from E to F is
Total number of onto function
= Total function from to – Number of function for which map and
Let \(g\left( x \right) = 1 + x – \left[ x \right]\) and \(f\left( x \right) = \left\{ \begin{array}{l} – 1,x 0\end{array} \right\}\), then fog(x) is equal to
is greater than .
Since
since
The function \(f:R \to \left[ { – \frac{1}{2},\frac{1}{2}} \right]\) defined as \(f(x) = \frac{x}{{1 + {x^2}}}\) is
or
is manyone.
Let
Range =Codomain =
So, is surjective.
Hence, is surjective but not injective.
The domain of the function \(f(x) = {\sin ^{ – 1}}\left( {{{\log }_2}\frac{{{x^2}}}{2}} \right)\) is
Domain of
If \({f_x}(x) = \frac{1}{k}({\sin ^k}x + {\cos ^k}x),x \in R,k \ge 1\) then \({f_4}(x) – {f_6}(x)\) is equal to
Let \(y(x)\) be a function defined on \([ – 1,1]\). If area of the equilateral triangle with two of its vertices at \((0,0)\) and \([x,y(x)]\) is \(\frac{{\sqrt 3 }}{4}\), then \(y(x)\) is
Area of equilateral triangle
Find the range of \(f(x) = \left {x – 9} \right\)
Range of \(\)f = [0, infty) /latex]
If \({[x]^2} – 11[x] + 18 = 0,\) then
Domain of \(f(x) = \sqrt {9 – {x^2}} \) is
is defined if
Domain of
If \(f(x) = ax + b,\) where \(ab \in z,f(3) = 3\) and \(f( – 5) = – 5\), then
and .
Solving the above equations
(i)
(ii)
a=2, b=3
Range of the function \(f(x) = \frac{{4 – x}}{{x – 4}}\) is
is defined if i.e.,
Let
where i.e.,
Range of
Domain of the function \(f(x) = \frac{{{x^2} + 2x + 1}}{{{x^2} – 5x – 6}}\) is
is defined if
Domain of