Find f+g, f-g, cf (c ∈ R, c ≠ 0), fg, 1/f and f/g in each of the following.
(i) \(f(x)=x^3+1\) and \(g(x)=x+1\)
1. Find \(f+g\)
\[
(f+g)(x)=f(x)+g(x)
\]
\[
=(x^3+1)+(x+1)
\]
\[
=x^3+x+2
\]
2. Find \(f-g\)
\[
(f-g)(x)=f(x)-g(x)
\]
\[
=(x^3+1)-(x+1)
\]
\[
=x^3-x
\]
3. Find \(cf\)
\[
(cf)(x)=c\cdot f(x)
\]
\[
=c(x^3+1)
\]
\[
=cx^3+c
\]
4. Find \(fg\)
\[
(fg)(x)=f(x)\cdot g(x)
\]
\[
=(x^3+1)(x+1)
\]
\[
=x^4+x^3+x+1
\]
5. Find \(1/f\)
\[
\left(\frac{1}{f}\right)(x)=\frac{1}{x^3+1}
\]
For reciprocal function, denominator should not be zero.
\[
x^3+1\neq 0
\]
\[
x\neq -1
\]
6. Find \(f/g\)
\[
\left(\frac{f}{g}\right)(x)=\frac{x^3+1}{x+1}
\]
\[
=\frac{(x+1)(x^2-x+1)}{x+1}
\]
\[
=x^2-x+1
\]
But denominator cannot be zero.
\[
x+1\neq 0
\]
\[
x\neq -1
\]
(ii) \(f(x)=\sqrt{x-1}\) and \(g(x)=\sqrt{x+1}\)
1. Find \(f+g\)
\[
(f+g)(x)=\sqrt{x-1}+\sqrt{x+1}
\]
Domain:
\[
x-1\geq0
\]
\[
x+1\geq0
\]
\[
x\geq1
\]
2. Find \(f-g\)
\[
(f-g)(x)=\sqrt{x-1}-\sqrt{x+1}
\]
Domain:
\[
x\geq1
\]
3. Find \(cf\)
\[
(cf)(x)=c\sqrt{x-1}
\]
Domain:
\[
x\geq1
\]
4. Find \(fg\)
\[
(fg)(x)=\sqrt{x-1}\cdot\sqrt{x+1}
\]
\[
=\sqrt{(x-1)(x+1)}
\]
\[
=\sqrt{x^2-1}
\]
Domain:
\[
x\geq1
\]
5. Find \(1/f\)
\[
\left(\frac{1}{f}\right)(x)=\frac{1}{\sqrt{x-1}}
\]
Denominator should not be zero.
\[
x-1>0
\]
\[
x>1
\]
6. Find \(f/g\)
\[
\left(\frac{f}{g}\right)(x)=\frac{\sqrt{x-1}}{\sqrt{x+1}}
\]
For domain:
\[
x-1\geq0
\]
\[
x+1>0
\]
\[
x\geq1
\]