Question

Find gof and fog when f : R → R and g : R → R are defined by
(i) f(x) = 2x + 3 and g(x) = x² + 5
(ii) f(x) = 2x + x² and g(x) = x³
(iii) f(x) = x² + 8 and g(x) = 3x³ + 1
(iv) f(x) = x and g(x) = |x|
(v) f(x) = x² + 2x − 3 and g(x) = 3x − 4
(vi) f(x) = 8x³ and g(x) = x^1/3

Answer 1

Given, f : R → R and g : R → R
So, gof : R → R and fog : R → R

(i) f(x) = 2x + 3 and g(x) = x² + 5
Now, (gof) (x)
= g (f (x))
= g (2x +3)
= (2x + 3)² + 5
= 4x²+ 9 + 12x +5
=4x²+ 12x + 14

(fog) (x)
=f (g (x))
= f (x² + 5)
= 2 (x² + 5) +3
= 2 x²+ 10 + 3
= 2x² + 13

(ii) f(x) = 2x + x² and g(x) = x^{3
$$\begin{gathered} \left(gof\right)\left(x\right) \\ =g\left(f\left(x\right)\right) \\ =g\left(2x+x^2\right) \\ ={\left(2x+x^2\right)}^3 \\ \left(fog\right)\left(x\right) \\ =f\left(g\left(x\right)\right) \\ =f\left(x^3\right) \\ =2\left(x^3\right)+{\left(x^3\right)}^2 \\ =2x^3+x^6 \end{gathered}$$}

(iii) f(x) = x² + 8 and g(x) = 3x³ + 1
$$\begin{gathered} \left(gof\right)\left(x\right) \\ =g\left(f\left(x\right)\right) \\ =g\left(x^2+8\right) \\ =3{\left(x^2+8\right)}^3+1 \\ \left(fog\right)\left(x\right) \\ =f\left(g\left(x\right)\right) \\ =f\left(3x^3+1\right) \\ ={\left(3x^3+1\right)}^2+8 \\ =9x^6+6x^3+1+8 \\ =9x^6+6x^3+9 \end{gathered}$$

(iv) f(x) = x and g(x) = |x|
$$\begin{gathered} \left(gof\right)\left(x\right) \\ =g\left(f\left(x\right)\right) \\ =g\left(x\right) \\ =\left|x\right| \\ \left(fog\right)\left(x\right) \\ =f\left(g\left(x\right)\right) \\ =f\left(\left|x\right|\right) \\ =\left|x\right| \end{gathered}$$

(v) f(x) = x² + 2x − 3 and g(x) = 3x − 4
$$\begin{gathered} \left(gof\right)\left(x\right) \\ =g\left(f\left(x\right)\right) \\ =g\left(x^2+2x-3\right) \\ =3\left(x^2+2x-3\right)-4 \\ =3x^2+6x-9-4 \\ =3x^2+6x-13 \\ \left(fog\right)\left(x\right) \\ =f\left(g\left(x\right)\right) \\ =f\left(3x-4\right) \\ ={\left(3x-4\right)}^2+2\left(3x-4\right)-3 \\ =9x^2+16-24x+6x-8-3 \\ =9x^2-18x+5 \end{gathered}$$

(vi) f(x) = 8x³ and g(x) = x^1/3
$$\begin{gathered} \left(gof\right)\left(x\right) \\ =g\left(f\left(x\right)\right) \\ =g\left(8x^3\right) \\ ={\left(8x^3\right)}^{\frac{1}{3}} \\ ={\left[{\left(2x\right)}^3\right]}^{\frac{1}{3}} \\ =2x \\ \left(fog\right)\left(x\right) \\ =f\left(g\left(x\right)\right) \\ =f\left(x^{\frac{1}{3}}\right) \\ =8{\left(x^{\frac{1}{3}}\right)}^3 \\ =8x \end{gathered}$$