Answer: [tex]f(x)=-x[/tex]
Step-by-step explanation:
When f(-x)= -f(x), then it is known as an odd function.
i) [tex]f(x) = x^3 + 5x^2 + x[/tex]
Then, [tex]f(-x)=(-x)^3+5(-x)^2+(-x)=-x^3+5x^2-x\neq -x^3-5x^2-x[/tex]
i.e. [tex]f(-x)\neq-f(x)[/tex]
ii) [tex]f(x)=\sqrt{x}[/tex]
[tex]f(-x)=\sqrt{-x}\neq-\sqrt{x}[/tex]
i.e. [tex]f(-x)\neq-f(x)[/tex]
iii) [tex]f(x)=x^2+x[/tex]
[tex]f(-x)=(-x)^2+(-x)=x^2-x\neq-x^2-x[/tex]
i.e. [tex]f(-x)\neq-f(x)[/tex]
iv) [tex]f(x)=-x[/tex]
[tex]f(-x)=-(-x)=x[/tex]
[tex]-f(x)=-(-x)=x[/tex]
i.e. [tex]f(-x)=-f(x)[/tex]
Hence, f(x) is an odd function.
