If f x - begin cases 2 x-3- - if x lt 0 1x- 2-1- -text if x 0lend cases and g x - begin cases 3 x-5- -text if x 1end cases -then the function fgx isA- one-one and ontoB- many one and ontoC- one-one and intoD- many one and into

The correct option is B many one and ontof(g(x))=2(g(x))+3 nbsp; amp; if g(x) 0 (g(x))^2 1 amp; if g(x) ≥ 0 = 2x^3+3 nbsp; amp;, x^3 0, n ...

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Byju's Answer

Standard XII

Mathematics

Test for Monotonicity about a Point

If f x =⟨ beg...

Question

If $f (x) = {\begin{matrix} 2 x + 3, & if x < 0 x^{2} - 1, & if x \geq 0 \end{matrix}$

and $g (x) = {\begin{matrix} x^{3}, & if x < 1 3 x - 5, & if x \geq 1 \end{matrix}$ ,

then the function $f (g (x))$ is

one-one and onto

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many one and onto

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one-one and into

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many one and into

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Solution

The correct option is B many one and onto
$f (g (x)) =$ ${\begin{matrix} 2 (g (x)) + 3 & if g (x) < 0 (g (x))^{2} - 1 & if g (x) \geq 0 \end{matrix}$

$= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 2 x^{3} + 3 & , x^{3} < 0, x < 1 2 (3 x - 5) + 3 & , 3 x - 5 < 0, x \geq 1 x^{6} - 1 & , x^{3} \geq 0, x < 1 (3 x - 5)^{2} - 1 & , 3 x - 5 \geq 0, x \geq 1 \end{matrix}$

$= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 2 x^{3} + 3 & , x \in (- \infty, 0) 6 x - 7 & , x \in [1, \frac{5}{3}) x^{6} - 1 & , x \in [0, 1) (3 x - 5)^{2} - 1 & , x \in [\frac{5}{3}, \infty) \end{matrix}$

$= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 2 x^{3} + 3 \in (- \infty, 3) 6 x - 7 \in [- 1, 3) x^{6} - 1 \in [- 1, 0) (3 x - 5)^{2} - 1 \in [- 1, \infty) \end{matrix}$

So, the range of $f (x)$ is $R$ and function is many-one.

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Similar questions

If $f (x) = {\begin{matrix} x, & if w h e n x i s r a t i o n a l 0, & if w h e n x i s i r r a t i o n a l \end{matrix}$ ;

$g (x) = {\begin{matrix} 0, & if w h e n x i s r a t i o n a l x, & if w h e n x i s i r r a t i o n a l \end{matrix}$ ;

then $(f - g$ ) is

If $f (x) =$ ${\begin{matrix} x, & when x is rational 0, & when x is irrational \end{matrix}$ ; $g (x) =$ ${\begin{matrix} 0, & when x is rational x, & when x is irrational \end{matrix}$ then $(f - g)$ is

Q. If

f (x) = {\begin{matrix} 2 x + 3, & if x < 0 x^{2} - 1, & if x \geq 0 \end{matrix}

and

g (x) = {\begin{matrix} x^{3}, & if x < 1 3 x - 5, & if x \geq 1 \end{matrix}

,

then the function

f (g (x))

K (x)

is a function such that

K (f (x)) = a + b + c + d

,
Where,
$a=\begin{cases}
0 & \text{ if f(x) is even} \\
-1 & \text{ if f(x) is odd} \\
2 & \text{ if f(x) is neither even nor odd}
\end{cases}$
$b=\begin{cases}
3 & \text{ if f(x) is periodic} \\
4 & \text{ if f(x) is aperiodic}
\end{cases}$
$c=\begin{cases}
5 & \text{ if f(x) is one one} \\
6 & \text{ if f(x) is many one}
\end{cases}$
$d=\begin{cases}
7 & \text{ if f(x) is onto} \\
8 & \text{ if f(x) is into}
\end{cases}$

h : R \to R, h (x) = (\frac{e^{2 x} + e^{x} + 1}{e^{2 x} - e^{x} + 1})

On the basis of above information, answer the following questions.

K (h (x)) =

K (x)

is a function such that

K (f (x)) = a + b + c + d

h : R \to R, h (x) = (\frac{e^{2 x} + e^{x} + 1}{e^{2 x} - e^{x} + 1})

On the basis of above information, answer the following questions.

K (ϕ (x))

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If f(x)={2x+3, if x<0x2−1, if x≥0 and g(x)={x3, if x<13x−5, if x≥1, then the function f(g(x)) is

If $f (x) = {\begin{matrix} 2 x + 3, & if x < 0 x^{2} - 1, & if x \geq 0 \end{matrix}$

and $g (x) = {\begin{matrix} x^{3}, & if x < 1 3 x - 5, & if x \geq 1 \end{matrix}$ ,

then the function $f (g (x))$ is