Question
Let $f(x)=\left\{\begin{array}{ll}x^2, & \text { when } 0 \leq x \leq 2 \\ 2 x, & \text { when } 2 \leq x \leq 5\end{array}\right.$
$
g(x)=\left\{\begin{array}{ll}
x^2, & \text { when } 0 \leq x \leq 3 \\
2 x, & \text { when } 3 \leq x \leq 5
\end{array}\right.
$
Show that $f$ is a function, while $g$ is not a function.

Answer

The relation $f$ is defined as
$
f(x)=\left\{\begin{array}{ll}
x^2, & 0 \leq x \leq 2 \\
2 x, & 2 \leq x \leq 5
\end{array}\right.
$
It is observed that for
$0 \leq x \leq 2, f(x)=x^2$
and $2 \leq x \leq 5, f(x)=2 x$
Also, at $x=2, f(x)=2^2=4$ or $f(x)=2 \times 2=4$
i.e., at $x=2, f(x)=4$.
Therefore, for $0 \leq x \leq 5$, the images of $f(x)$ are unique. Thus, the given relation ' $f$ ' is a function.
The relation g is defined as $g(x)=\left\{\begin{array}{ll}x^2, & 0 \leq x \leq 3 \\ 2 x, & 3 \leq x \leq 5\end{array}\right.$
It can be observed that for $x=3, g(x)=3^2=9$ and $g(x)=2 \times 3=6$
Hence, element 3 of the domain of relation ' $g$ ' corresponds to two different images i.e., 9 and 6 . Hence, this relation is not a function.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Similar questions

If $y=a e^{2 x}+b e^{-x}$, then show that $\frac{d^2 y}{d x^2}-\frac{d y}{d x}-2 y=0$.
A bag contains 8 red and 5 white balls. Two successive draws of all 3 balls are made at random from the bag without replacements. Find the probability that the first draw yields 3 white balls and second draw yields 3 red balls.
In an A.P., if the $p$ th term is $\frac{1}{q}$ and $q$ th term is $\frac{1}{p}$.
Prove that the sum of first $p q$ term is $\frac{1}{2}(p q+1)$.
A manufacturer in a firm manufactures a machine and marks it at ₹ $80,000.$ He sells the machine to a wholesaler (in Gorakhpur) at a discount of $20 \%$ The wholesaler sells the machine to a dealer (in Mathura) at a discount of $15 \%$ on the marked price. If the rate of GST $28 \%,$ find tax paid by the whole saler to central Government.
If $P(E)=\frac{7}{13}, P(F)=\frac{9}{13}$ and $P(E \cap F)=\frac{4}{13}$, then evaluate: (i) $P\left(\frac{\bar{E}}{F}\right)$ (ii) $P\left(\frac{\bar{E}}{\bar{F}}\right)$.
'A' earns the following income during the financial year 2018-19 :
ParticularsAmount (₹)
(a) Interest paid by an Indian company but received in London2,00,000
(b) Pension from former employer in India, received in USA8,000
(c) Profits earned from business in Paris which is controlled in India, half of the profits being received in India40,000
(d) Income from agriculture in Bhutan and remitted to India10,000
(e) Income from property in England and received there8,000
(f) Past foreign untaxed income brought to India20,000
Determine the total income of 'A' for the assessment year 2019-20 if he is (i) Resident and ordinarily resident, (ii) Not ordinarily resident, and (iii) Non-resident in India.
The ratio of the sum of $n$ terms of two A.P.'s is $(7 n-1):(3 n+11)$. Find the ratio of their $10^{\text {th }}$ terms.
Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
If $y=\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}}$, prove that $(2 x y) \frac{d y}{d x}=\frac{x}{a}-\frac{a}{x}$.
The circumference of a circular garden is 1012 m . Find the area of outsides road of 3.5 m width runs around it. Calculate the area of this road and find the cost of gravelling the road at ₹ 32 per 100 sq.m.