1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

# Find fog and gof if (i) $f\left(x\right)={e}^{x},g\left(x\right)={\mathrm{log}}_{e}x$ (ii) $f\left(x\right)={x}^{2},g\left(x\right)=\mathrm{cos}x$ (iii) $f\left(x\right)=|x|,g\left(x\right)=\mathrm{sin}x$ (iv) $f\left(x\right)=x+1,g\left(x\right)={e}^{x}$ (v) $f\left(x\right)={\mathrm{sin}}^{-1}x,g\left(x\right)={x}^{2}$ (vi) $f\left(x\right)=x+1,g\left(x\right)=\mathrm{sin}x$ (vii) $f\left(x\right)=x+1,g\left(x\right)=2x+3$ (viii) $f\left(x\right)=c,c\in R,g\left(x\right)=\mathrm{sin}{x}^{2}$ (ix) $f\left(x\right)={x}^{2}+2,g\left(x\right)=1-\frac{1}{1-x}$

Open in App
Solution

## $\left(i\right)f\left(x\right)={e}^{x},g\left(x\right)={\mathrm{log}}_{e}x\phantom{\rule{0ex}{0ex}}f:R\to \left(0,\infty \right);g:\left(0,\infty \right)\to R\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}fog\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}g\text{is a subset of the domain of}f.\phantom{\rule{0ex}{0ex}}fog:\left(0,\infty \right)\to R\phantom{\rule{0ex}{0ex}}\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=f\left({\mathrm{log}}_{e}x\right)\phantom{\rule{0ex}{0ex}}={\mathrm{log}}_{e}{e}^{x}\phantom{\rule{0ex}{0ex}}=x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}gof\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}f\text{is a subset of the domain of}g.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=g\left({e}^{x}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{log}}_{e}{e}^{x}\phantom{\rule{0ex}{0ex}}=x$ $\left(ii\right)f\left(x\right)={x}^{2},g\left(x\right)=\mathrm{cos}x\phantom{\rule{0ex}{0ex}}f:R\to \left[0,\infty \right);g:R\to \left[-1,1\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}\mathrm{fog}\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}g\text{is not a subset of the domain of}f.\phantom{\rule{0ex}{0ex}}⇒\text{Domain}\left(fog\right)\text{=}\left\{x:x\in \text{domain of}g\text{and}g\left(x\right)\in \text{domain of}f\right\}\phantom{\rule{0ex}{0ex}}⇒\text{Domain}\left(fog\right)\text{=}x:x\in R\text{and}\mathrm{cos}x\in R\right\}\phantom{\rule{0ex}{0ex}}⇒\text{Domain of}\left(fog\right)\text{=}R\phantom{\rule{0ex}{0ex}}fog:R\to R\phantom{\rule{0ex}{0ex}}\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=f\left(\mathrm{cos}x\right)\phantom{\rule{0ex}{0ex}}={\mathrm{cos}}^{2}x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}gof\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}f\text{is a subset of the domain of}g.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=g\left({x}^{2}\right)\phantom{\rule{0ex}{0ex}}=\mathrm{cos}\left({x}^{2}\right)$ $\left(iii\right)f\left(x\right)=\left|x\right|,g\left(x\right)=\mathrm{sin}x\phantom{\rule{0ex}{0ex}}f:R\to \left(0,\infty \right);g:R\to \left[-1,1\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}fog\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}g\text{is a subset of the domain of}f.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=f\left(\mathrm{sin}x\right)\phantom{\rule{0ex}{0ex}}=\left|\mathrm{sin}x\right|\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}gof\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}f\text{is a subset of the domain of}g.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=g\left(\left|x\right|\right)\phantom{\rule{0ex}{0ex}}=\mathrm{sin}\left|x\right|$ $\left(iv\right)f\left(x\right)=x+1,g\left(x\right)={e}^{x}\phantom{\rule{0ex}{0ex}}f:R\to R;g:R\to \left[1,\infty \right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}fog\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, range of}g\text{is a subset of domain of}f.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=f\left({e}^{x}\right)\phantom{\rule{0ex}{0ex}}={e}^{x}+1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}gof\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, range of}f\text{is a subset of domain of}g.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=g\left(x+1\right)\phantom{\rule{0ex}{0ex}}={e}^{x+1}$ $\left(v\right)f\left(x\right)={\mathrm{sin}}^{-1}x,g\left(x\right)={x}^{2}\phantom{\rule{0ex}{0ex}}f:\left[-1,1\right]\to \left[\frac{-\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right];g:R\to \left[0,\infty \right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathbf{\text{Computing}}\mathbit{f}\mathbit{o}\mathbit{g}\mathbf{\text{:}}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}g\text{is not a subset of the domain of}f.\phantom{\rule{0ex}{0ex}}\text{Domain}\left(fog\right)\text{=}\left\{x:x\in \text{domain of}g\text{and}g\left(x\right)\in \text{domain of}f\right\}\phantom{\rule{0ex}{0ex}}\text{Domain}\left(fog\right)\text{=}\left\{x:x\in R\text{and}{x}^{2}\in \left[-1,1\right]\right\}\phantom{\rule{0ex}{0ex}}\text{Domain}\left(fog\right)\text{=}\left\{x:x\in R\text{and}x\in \left[-1,1\right]\right\}\phantom{\rule{0ex}{0ex}}\text{Domain of}\left(fog\right)\text{=}\left[-1,1\right]\phantom{\rule{0ex}{0ex}}fog:\left[-1,1\right]\to R\phantom{\rule{0ex}{0ex}}\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=f\left({x}^{2}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{sin}}^{-1}\left({x}^{2}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathbf{\text{Computing}}\mathbit{g}\mathbit{o}\mathbit{f}\mathbf{\text{:}}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}f\text{is a subset of the domain of}g.\phantom{\rule{0ex}{0ex}}fog:\left[-1,1\right]\to R\phantom{\rule{0ex}{0ex}}\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=g\left({\mathrm{sin}}^{-1}x\right)\phantom{\rule{0ex}{0ex}}={\left({\mathrm{sin}}^{-1}x\right)}^{2}$ $\left(\mathrm{vi}\right)f\left(x\right)=x+1,g\left(x\right)=\mathrm{sin}x\phantom{\rule{0ex}{0ex}}f:R\to R;g:R\to \left[-1,1\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}fog\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}g\text{is a subset of the domain of}f.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=f\left(\mathrm{sin}x\right)\phantom{\rule{0ex}{0ex}}=\mathrm{sin}x+1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}gof\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}f\text{is a subset of the domain of}g.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=g\left(x+1\right)\phantom{\rule{0ex}{0ex}}=\mathrm{sin}\left(x+1\right)$ $\left(vii\right)f\left(x\right)=x+1,g\left(x\right)=2x+3\phantom{\rule{0ex}{0ex}}f:R\to R;g:R\to R\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}fog\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}g\text{is a subset of the domain of}f.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=f\left(2x+3\right)\phantom{\rule{0ex}{0ex}}=2x+3+1\phantom{\rule{0ex}{0ex}}=2x+4\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}gof\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}f\text{is a subset of the domain of}g.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=g\left(x+1\right)\phantom{\rule{0ex}{0ex}}=2\left(x+1\right)+3\phantom{\rule{0ex}{0ex}}=2x+5$ $\left(viii\right)f\left(x\right)=c,g\left(x\right)=\mathrm{sin}{x}^{2}\phantom{\rule{0ex}{0ex}}f:R\to \left\{c\right\};g:R\to \left[0,1\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}fog\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}g\text{is a subset of the domain of}f.\phantom{\rule{0ex}{0ex}}fog:R\to R\phantom{\rule{0ex}{0ex}}\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=f\left(\mathrm{sin}{x}^{2}\right)\phantom{\rule{0ex}{0ex}}=c\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}gof\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}f\text{is a subset of the domain of}g.\phantom{\rule{0ex}{0ex}}⇒fog:R\to R\phantom{\rule{0ex}{0ex}}\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=g\left(c\right)\phantom{\rule{0ex}{0ex}}=\mathrm{sin}{c}^{2}$ $\left(ix\right)f\left(x\right)={x}^{2}+2\phantom{\rule{0ex}{0ex}}f:R\to \left[2,\infty \right)\phantom{\rule{0ex}{0ex}}g\left(x\right)=1-\frac{1}{1-x}\phantom{\rule{0ex}{0ex}}\text{For domain of}g:1-x\ne 0\phantom{\rule{0ex}{0ex}}⇒x\ne 1\phantom{\rule{0ex}{0ex}}⇒\text{Domain of g=}R-\left\{1\right\}\phantom{\rule{0ex}{0ex}}g\left(x\right)=1-\frac{1}{1-x}=\frac{1-x-1}{1-x}=\frac{-x}{1-x}\phantom{\rule{0ex}{0ex}}\text{For range of}g:\phantom{\rule{0ex}{0ex}}y=\frac{-x}{1-x}\phantom{\rule{0ex}{0ex}}⇒y-xy=-x\phantom{\rule{0ex}{0ex}}⇒y=xy-x\phantom{\rule{0ex}{0ex}}⇒y=x\left(y-1\right)\phantom{\rule{0ex}{0ex}}⇒x=\frac{y}{y-1}\phantom{\rule{0ex}{0ex}}\text{Range of g =}R-\left\{1\right\}\phantom{\rule{0ex}{0ex}}\text{So, g:}R-\left\{1\right\}\to R-\left\{1\right\}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}fog\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}g\text{is a subset of the domain of}f.\phantom{\rule{0ex}{0ex}}⇒fog:R-\left\{1\right\}\to R\phantom{\rule{0ex}{0ex}}\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=f\left(\frac{-x}{x-1}\right)\phantom{\rule{0ex}{0ex}}={\left(\frac{-x}{x-1}\right)}^{2}+2\phantom{\rule{0ex}{0ex}}=\frac{{x}^{2}+2{x}^{2}+2-4x}{{\left(1-x\right)}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{3{x}^{2}-4x+2}{{\left(1-x\right)}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Computing}gof\text{:}\phantom{\rule{0ex}{0ex}}\text{Clearly, the range of}f\text{is a subset of the domain of}g.\phantom{\rule{0ex}{0ex}}⇒gof:R\to R\phantom{\rule{0ex}{0ex}}\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)\phantom{\rule{0ex}{0ex}}=g\left({x}^{2}+2\right)\phantom{\rule{0ex}{0ex}}=1-\frac{1}{1-\left({x}^{2}+2\right)}\phantom{\rule{0ex}{0ex}}=1-\frac{1}{-\left({x}^{2}+1\right)}\phantom{\rule{0ex}{0ex}}=\frac{{x}^{2}+2}{{x}^{2}+1}$

Suggest Corrections
1
Join BYJU'S Learning Program
Related Videos
Extrema
MATHEMATICS
Watch in App
Join BYJU'S Learning Program