Continuity of Composite Functions

Q.

Explain the term entire function.

Q. If $f\left(x\right)=2x$ and $g\left(x\right)=\frac{x^2}{2}+1$ , then which of the following can be a discontinuous function

1. $f\left(x\right)+g\left(x\right)$
2. $f\left(x\right)–g\left(x\right)$
3. $f\left(x\right).g\left(x\right)$
4. $\frac{g\left(x\right)}{f\left(x\right)}$
5. $\frac{f\left(x\right)}{g\left(x\right)}$

Q.

Show that the function f Defined by f(x)=|1-x+|x|| is a continuous function

Q. If f(x) is continuous and g(x) is discontinuous at x = a, then the product function h(x) = f(x)g(x) must be discontinuous at x = a.

1. False
2. True

Q. In expansion of (x-1)(x-2)....(x-100) what will be the coefficient of x^99 ?

Q. The composition of two continuous function is a continuous function

1. True
2. False

Q.

Integral( 2x^12 +5x^9)/(x^5+x^3+1)^3

Q. The function defined by $f\left(x\right)=(-1{)}^{\left[x^3\right]}$ , where [ ] denotes greatest integer function satisfies

1. Continuous at all $x\in Z$
2. $f(\frac{3}{2}=1)$
3. Discontinuous for $x=n^{\frac{1}{3}},n\in Z$
4. $f^{\prime }\left(x\right)=0for-1<x<1$

Q. If $f\left(x\right)=2x$ and $g\left(x\right)=\frac{x^2}{2}+1$ , then which of the following can be a discontinuous function

1. $f\left(x\right)+g\left(x\right)$
2. $f\left(x\right)–g\left(x\right)$
3. $f\left(x\right).g\left(x\right)$
4. $\frac{g\left(x\right)}{f\left(x\right)}$
5. $\frac{f\left(x\right)}{g\left(x\right)}$

Q. Let $f:R\to R$ and $g:R\to R$ be defined as
$f\left(x\right)=\{\begin{array}{cc}x+a,& x<0\\ |x-1|,& x\ge 0\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}x+1,& x<0\\ (x-1{)}^2+b,& x\ge 0\end{array},$
where $a,b$ are non-negative real numbers. If $\left(gof\right)\left(x\right)$ is continuous for all $x\in R,$ then $a+b$ is equal to

Q. Find the derivative of the function f defined by f (x) = mx + c at x = 0.

Q. Show that the derivative of the function f given by
$f\left(x\right)=2x^3-9x^2+12x+9$, at x = 1 and x = 2 are equal.

Q. The degree of the differential equation ${\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^2+\sin \left(\frac{dy}{dx}\right)+1=0$, is
(a) 3
(b) 2
(c) 1
(d) not defined

Q. If f(x) is continuous and g(x) is discontinuous at x = a, then the product function h(x) = f(x)g(x) must be discontinuous at x = a.

1. False
2. True

Q. If $f\left(x\right)=\frac{1}{(x-1)(x-2)}andg\left(x\right)=\frac{1}{x^2}$, then points of discontinuity of f(g(x)) are

1. $\{-1,0,1,\frac{1}{\sqrt{2}}\}$
2. $\{-\frac{1}{\sqrt{2}},-1,0,1,\frac{1}{\sqrt{2}}\}$
3. $\{0,1\}$
4. $\{0,1,\frac{1}{\sqrt{2}}\}$

Q.

(ay³+by^​3/2+c) is it a quadratic expression in y^{​​​​​​3/2}

Q. Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.

Q. Let f(x) be a polynomial. Then, the second order derivative of f(e^x) is

(a) f'' (e^x) e^2x + f'(e^x) e^x
(b) f'' (e^x) e^x + f' (e^x)
(c) f'' (e^x) e^2x + f'' (e^x) e^x
(d) f'' (e^x)

Q. The set of points where the function f (x) = x |x| is differentiable is
(a) $\left(-\infty ,\infty \right)$
(b) $\left(-\infty ,0\right)\cup \left(0,\infty \right)$
(c) $\left(0,\infty \right)$
(d) $\left[0,\infty \right]$

Q. ntIf f(x) be q polynomial of degree 4, such that f(0)=0, f(-1)=55 and f(x) has the points of relative maximum or relative minimum at x=1, 2, 3 thenn ntA) f(x) has two points of relative minima and one point of relative Maxima.n ntB) range of f(x) contains 9 negative integersn ntC) sum of real roots of f(x)=0 is 4n ntD) f(x) has exactly one inflection point.n ntn nt(One or more than one options correct type)n

Q. The points of discontinuity of the function
$f\left(x\right)=\left\{\begin{array}{cc}\frac{1}{5}\left(2x^2+3\right),& x\le 1\\ 6-5x,& 1<x<3\\ x-3,& x\ge 3\end{array}\right.\mathrm{is}\left(\mathrm{are}\right)$
(a) x = 1
(b) x = 3
(c) x = 1, 3
(d) none of these

Q.

If f(x) = |x - 2|, then at x = 2, f'(x) is

1. Neither continous nor differentiable

2. Continous but not differentiable

3. Continous and differentiable both

4. Differentiable but not continous

Q. Let f(x) be a polynomial. Then, the second order derivative of f(e^x) is

(a) f'' (e^x) e^2x + f'(e^x) e^x
(b) f'' (e^x) e^x + f' (e^x)
(c) f'' (e^x) e^2x + f'' (e^x) e^x
(d) f'' (e^x)

Q. If f is defined by $f\left(x\right)=x^2-4x+7$, show that $f\text{'}\left(5\right)=2f\text{'}\left(\frac{7}{2}\right)$

Q. If $f\left(x\right)=2x$ and $g\left(x\right)=\frac{x^2}{2}+1$ , then which of the following can be a discontinuous function

1. $f\left(x\right)+g\left(x\right)$
2. $f\left(x\right)–g\left(x\right)$
3. $f\left(x\right).g\left(x\right)$
4. $\frac{g\left(x\right)}{f\left(x\right)}$
5. $\frac{f\left(x\right)}{g\left(x\right)}$

Q. If f : R → R be defined by f(x) = x³ −3, then prove that f⁻¹ exists and find a formula for f⁻¹. Hence, find f⁻¹ (24) and f⁻¹ (5).

Q. The composition of two continuous function is a continuous function

1. True
2. False

Q. If $f\left(x\right)=x^3+7x^2+8x-9$, find f'(4).