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Classify the following function as injection, surjection or bijection: f: R → R, defined by

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{x}~+\text{ }\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}\mathbf{x}\]

CBSE | Class 12 | Exercise 2.1 | Function | Maths | RD Sharma
Classify the following function as injection, surjection or bijection: f: R → R, defined by

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{x}~+\text{ }\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}\mathbf{x}\]

Given f: R → R, defined by 

    \[f\left( x \right)\text{ }=\text{ }si{{n}^{2}}x~+\text{ }co{{s}^{2}}x\]

Now we have to check for the given function is injection, surjection and bijection condition.

Injection condition:

    \[f\left( x \right)\text{ }=\text{ }si{{n}^{2}}x~+\text{ }co{{s}^{2}}x\]

We know that

    \[si{{n}^{2}}x~+\text{ }co{{s}^{2}}x~=\text{ }1\]

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.

Surjection condition:

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.

i

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