 # Algebra of Real Function

## Algebra of Real Function

Real Value Function: A function which has all real number or subset of the real number as it domain
Real Valued Function: A function which has all real number or subset of the real number as it range
For functions $f:X - > {\bf{R}}$ and $g:X - > {\bf{R}}$, we have
$\left( {f + g} \right)\left( x \right) = f\left( x \right) + g\left( x \right),x \in X$
2. Substraction
$\left( {f - g} \right)\left( x \right) = f\left( x \right)-g\left( x \right),x \in X$
3. Multiplication
$\left( {f.g} \right)\left( x \right) = f\left( x \right).g\left( x \right),x \in X$
4. Multiplication by real number $\left( {kf} \right)\left( x \right) = kf\left( x \right),x \in X$, where $k$ is a real number.
5. Division
$\frac{f}{g}\left( x \right) = \frac{{f(x)}}{{g(x)}}$
$x \in X$ and $g\left( x \right) \ne 0$

### Quiz Time

Question 1.
which is of the below relation is not a function.
A) R = {(2,2),(2,4),(3,3), (4,4)}
B) R = {(2,1),(3,1), (4,2)}
C) R = {(1,2),(2,3),(3,4), (4,5), (5,6), (6,7)}
D) None of the above
Question 2.
Let A = {1, 2, 3,4} and B = {5, 7}. Then possible number of relation from A to B ?
A) 64
B) 256
C) 16
D) 32
Question 3.
Let p(x) = x3 and q(x) = 2x + 1 be two real functions,then which of the following is incorrect
A) $(f + g) (x) = x^3 + 2x + 1$
B) $(f -g) (x) = x^3- 2x - 1$
C) $(fg) (x) = 2x^3 +1$
D) $\frac {f}{g}x=\frac {x^3}{2x+1} ,x \neq \frac {1}{2}$
Question 4.
Let f(x) =x2 ,find the value of $\frac {f(2.1) -f(2)}{2.1 -2}$
A) 4
B) .41
C) 4.1
D) .14
Question 5.
Find the range of the function defined as $f(x)=\sqrt {9-x^2}$
A) [-3,3]
B) [0,3]
C)(0,3)
D)[-3,0)
Question 6.
Find the domain of the function $f(x) =\frac {x^2+1}{x^2-5x+4}$
A) R
B) R - {4}
C) R - {1}
D) R -{1,4}