Relations and Functions Test

Result

Relations and Functions Test - 65

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Weekly Quiz Competition

Question 1
1 / -0

If $$f\left( x \right) =\begin{cases} 2+x,\quad x\ge 0 \\ 2-x,\quad x<0 \end{cases}$$ then $$f\left( f\left( x \right) \right) $$ is given by

A
$$f\left( f\left( x \right) \right) =\begin{cases} 2+x,\quad x\ge 0 \\ 4-x,\quad x<0 \end{cases}$$

B
$$f\left( f\left( x \right) \right) =\begin{cases} 2+x,\quad x\ge 0 \\ 2-x,\quad x<0 \end{cases}$$

C
$$f\left( f\left( x \right) \right) =\begin{cases} 4+x,\quad x< 0 \\ x,\quad x\ge 0 \end{cases}$$

D
$$f\left( f\left( x \right) \right) =\begin{cases} 4+x,\quad x\ge 0 \\ x,\quad x<0 \end{cases}$$

Solution
Question 2
1 / -0

Let $$f:X \to \left[ {1,\,27} \right]$$ be a function by $$f\left( x \right) = 5\sin x + 12\cos x + 14$$. The set $$X$$ so that $$f$$ is one-one and onto is

A
$$\left[ { - \pi /2,\pi /2\,} \right]$$

B
$$\left[ {0,\,\pi } \right]$$

C
$$\left[ {0,\,\pi /2} \right]$$

D
non of these

Solution
Question 3
1 / -0

If : $$f(x) = 5 {x}^{2}$$, $$g(x) = 3x^{4}$$, then : $$(fog) (-1) =$$

A
$$45$$

B
$$-54$$

C
$$-32$$

D
$$-64$$

Solution
Question 4
1 / -0

For $$a,\ b\ \in \ R-\left\{ 0 \right\}$$, let $$f(x)=ax^{2}+bx+a$$ satisfies $$f\left(x+\dfrac{7}{4}\right)=f\left(\dfrac{7}{4}-x\right) \forall \ x\ \in\ R$$.
Also the equation $$f(x)=7x+a$$ has only one real distinct solution. The minimum value of $$f(x)$$ in $$\left[0,\dfrac{3}{2}\right]$$ is equal to

A
$$\dfrac{-33}{8}$$

B
$$0$$

C
$$4$$

D
$$-2$$

Solution
Question 5
1 / -0

If $$f\left( x \right) = (1 - x)$$ , $$x \in \left[ { - 3,3} \right]$$ , then the domain of $$f\left( {f\left( x \right)} \right)$$ is

A
$$\left[ { - 2,3} \right]$$

B
$$\left( { - 2,3} \right)$$

C
$$\left[ { - 2,3]} \right.$$

D
$$( - 2,3]$$

Solution

$$f\left(x\right)=1-x$$
$$f\left(f\left(x\right)\right)=f\left(1-x\right)=1-1+x=x$$
And $$x\in\left[-3,3\right]$$
Domain of $$f\left(x\right)$$ is $$\left[-3,3\right]$$
Domain of $$f\left(f\left(x\right)\right)=$$Domain of $$f\left(1-x\right)$$
Domain of $$f\left(f\left(x\right)\right)=\left[1-3,3\right]=\left[-2,3\right]$$
Question 6
1 / -0

If f(g(x))=5x+2 and g(x)=8x then f(x)=

A
$$ \frac{5}{8}x+2$$.

B
$$ \frac{8}{5}x+2$$.

C
$$ \frac{5}{8}x-2$$.

D
8x-2

E
5x-2

Solution
Question 7
1 / -0

Let $$g\left( x \right) =1+x-\left[ x \right] $$ and $$f\left( x \right) =\begin{cases} -1,x<0 \\ 0,x=0 \\ 1,x>0 \end{cases}$$ Then for all $$x,f\left( g\left( x \right) \right) $$ is equal to (where $$\left[ . \right] $$ represents the greatest integer function)

A
$$x$$

B
$$1$$

C
$$f\left( x \right)$$

D
$$g\left( x \right)$$

Solution
Question 8
1 / -0

If $$f:R \rightarrow R, f(x)=2x-1$$ and $$g; R \rightarrow R, g(x)=x^{2}+2$$, then $$(gof)(x)$$ equals-

A
$$2x^{2}-1$$

B
$$(2x-1)^{2}$$

C
$$2x^{2}+3$$

D
$$4x^{2}-4x+3$$

Solution

$$f(x)=2x-1$$
$$g(x)=x^2+2$$
$$g(f(x))=(2x-1)^2+2=4x^2-4x+3$$.
Question 9
1 / -0

Let $$g(x)=1+x-[x]\quad $$ and $$f(x)=\begin{cases} -1\quad if\quad x<0 \\ 0\quad \quad if\quad x=0 \\ 1\quad \quad if\quad x>0 \end{cases}$$ , then $$\forall \:x,fog(x)$$ equals

A
$$x$$

B
$$1$$

C
$$f(x)$$

D
$$g(x)$$

Solution

given g(x)=1+x=[x] 1+{x} {x} donates the fractional part of x so,
g(x) is always positive
so, input to f(g(x)) is always positive and
hence, fog(x)=1
Question 10
1 / -0

The last three digits, if $$(12345956)_{10}$$ is expressed in binary system.

A
$$110$$

B
$$210$$

C
$$100$$

D
$$010$$

Solution

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