Classify the following functions as injection, surjection or bije… - Vidyadip

Question

Classify the following functions as injection, surjection or bijection:
$f : R \rightarrow R,$ defined by $f(x) = \sin^2x + \cos^2x$

✓

Answer

$f : R \rightarrow R,$ defined by $f(x) = \sin^2x + \cos^2x$
$f(x) = \sin^2x + \cos^2x = 1$
So, $f(x) = 1$ for every $x$ in $R.$
So, for all elements in the domain, the image is $1.$
So, $f$ is not an injection.
Range of $f = \{1\}$
Co-domain of $f = R$
Both are not same.
So, $f$ is not a surjection and $f$ is not a bijection.

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

Explore more

More "Solve the Following Question.(3 Marks)" from Functions→Functions — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Let $f : R → R$ and $g : R → R$ be defined by $f(x) = x + 1$ and $g(x) = x - 1$. Show that $fog = gof = I_R.$

Find the mean and standard deviation of the following probability distributions:

$x_i$	2	3	4
$p_i$	0.2	0.5	0.3

Solve the following differential equations:$\text{ye}^{\frac{\text{x}}{\text{y}}}\text{dx}=(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}^2)\text{dy, y}\neq0$

Find a vector of magnitude of 5 units parallel to the resultant of the vectors $\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$.

Evaluate the following definite integrals:$\int_{0}^\limits{1}\frac{1}{\sqrt{1+\text{x}}-\sqrt{\text{x}}}\text{ dx}$

If $\text{y}=\sin\text{x}$ and x changes from $\frac{\pi}{2}$ to $\frac{22}{14}$, what is the approximate change in y?

Write the following in the simplest form:
$\sin^{-1}\Big\{\frac{\sqrt{1+\text{x}}+\sqrt{1-\text{x}}}{2}\Big\},0<\text{x}<1$

Classify the following functions as injection, surjection or bijection:
f : Z → Z, defined by f(x) = x - 5

If $\text{A}=\begin{bmatrix}4 & 5 \\2 & 1 \end{bmatrix},$ then show that $A - 3I = 2 (I + 3A^{-1})$.

Find the mean and standard deviation of the following probability distributions:

$x_i$	-2	-1	0	1	2
$p_i$	0.1	0.2	0.4	0.2	0.1