Login/Sign Up
Home » If f(x) =     and g(x) =     be two real functions, then describe each of the following functions: (i) fog (ii) gof (iii) fof (iv)     Also, show that    
If f(x) =

    \[\mathbf{2x}~+\text{ }\mathbf{5}\]

and g(x) =

    \[{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{1}\]

be two real functions, then describe each of the following functions: (i) fog (ii) gof (iii) fof (iv)

    \[{{\mathbf{f}}^{\mathbf{2}}}\]

Also, show that

    \[\mathbf{fof}~\ne ~{{\mathbf{f}}^{\mathbf{2}}}\]

CBSE | Class 12 | Exercise 2.3 | Function | Maths | RD Sharma
If f(x) =

    \[\mathbf{2x}~+\text{ }\mathbf{5}\]

and g(x) =

    \[{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{1}\]

be two real functions, then describe each of the following functions: (i) fog (ii) gof (iii) fof (iv)

    \[{{\mathbf{f}}^{\mathbf{2}}}\]

Also, show that

    \[\mathbf{fof}~\ne ~{{\mathbf{f}}^{\mathbf{2}}}\]

f(x) and g(x) are polynomials.

⇒ f: R → R and g: R → R.

So, fog: R → R and gof: R → R.

(i) (fog) (x) = f (g (x))

=

    \[f\text{ }({{x}^{2}}~+\text{ }1)\]

=

    \[2\text{ }({{x}^{2~}}+\text{ }1)\text{ }+\text{ }5\]

=

    \[2{{x}^{2}}~+\text{ }2\text{ }+\text{ }5\]

=

    \[2{{x}^{2}}~+7\]

(ii) (gof) (x) = g (f (x))

=

    \[g\text{ }\left( 2x\text{ }+5 \right)\]

=

    \[{{\left( 2x\text{ }+\text{ }5 \right)}^{2}}~+\text{ }1\]

=

    \[4{{x}^{2}}~+\text{ }20x\text{ }+\text{ }26\]

(iii) (fof) (x) = f (f (x))

=

    \[f\text{ }\left( 2x\text{ }+5 \right)\]

=

    \[2\text{ }\left( 2x\text{ }+\text{ }5 \right)\text{ }+\text{ }5\]

=

    \[4x\text{ }+\text{ }10\text{ }+\text{ }5\]

=

    \[4x\text{ }+\text{ }15\]

(iv) f2 (x) = f (x) x f (x)

=

    \[\left( 2x\text{ }+\text{ }5 \right)\text{ }\left( 2x\text{ }+\text{ }5 \right)\]

=

    \[{{\left( 2x\text{ }+\text{ }5 \right)}^{2}}\]

=

    \[4{{x}^{2}}~+\text{ }20x\text{ }+25\]

Hence, from (iii) and (iv) clearly

    \[fof~\ne ~{{f}^{2}}\]

i

More Material