Let $\text{f(x)}=\sqrt{\text{x}}$ and $\text{g(x)}=\text{x}$ be two functions defined in the domain $\text{R}^+\cup\{0\}$. Find:
$\text{(fg)(x)}$
Answer
We have, $\text{f(x)}=\sqrt{\text{x}}$ and $\text{g(x)}=\text{x}$ be two functions defined in the domain $\text{R}^+\cup\{0\}.$
$(\text{f}\text{g})(\text{x})=\text{f(x)}.\text{g(x)}=\sqrt{\text{x}}.\text{x}=\text{x}^{\frac{3}{2}}$
Is the given relation a function? Give reasons for your answe $r:s = (n, n^{2 }) | n$ is a positive integer
Answer
We have$, s = (n, n^2) |$ n is a positive integer
Since the square of any positive integer is unique$,$ every element in the domain has unique image. Hence$, S$ is a function.
Is the given relation a function? Give reasons for your answer: $\text{g}=\big\{\text{n}, \frac{1}{\text{n}}\big\},$ | n is a positive integer
Answer
We have, $\text{g}=\big\{\text{n}, \frac{1}{\text{n}}\big\},$ | n is a positive integer} For n, it is a positive integer and $\frac{1}{\text{n}}$ is unique and distinct. Therefore,every element in the domain has unique image. So, it is a function.
Given $\text{A} = \big\{1, 2, 3, 4, 5\big\}$$\text{S} = \text{(x, y)} : \text{x} \in\text{A}, \text{y} \in\text{A}.$ Find the ordered pairs which satisfy the condition given below: X + Y = 5
Answer
Given that: $\text{A} = \big\{1, 2, 3, 4, 5\big\}$ $\text{S} = \text{(x, y)} : \text{x} \in\text{A}, \text{y} \in\text{A}.$ X + Y = 5, So, the ordered pairs satisfying the given conditions are (1, 4), (4, 1), (2, 3), (3, 2).
Find the range of the following functions given by:
$\text{f(x)}=1+3\cos2\text{x}$
Answer
We knoe that, $-1\leq\cos2\text{x}\leq1$
$\Rightarrow-3\leq3\cos2\text{x}\leq3$
$\Rightarrow-2\leq1+3\cos2\text{x}\leq4$
$\Rightarrow-2\leq\text{f(x)}\leq4$
$\therefore $ Range of $\text{f}=[-2, 4]$
Given $\text{A} = \big\{1, 2, 3, 4, 5\big\}$$\text{S} = \text{(x, y)} : \text{x} \in\text{A}, \text{y} \in\text{A}.$ Find the ordered pairs which satisfy the condition given below: X + Y < 5
Answer
Given that: $\text{A} = \big\{1, 2, 3, 4, 5\big\}$ $\text{S} = \text{(x, y)} : \text{x} \in\text{A}, \text{y} \in\text{A}.$ X + Y < 5, So, the ordered pairs satisfying the given conditions are (1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1).
Let $\text{f(x)}=\sqrt{\text{x}}$ and $\text {g(x)}=\text{x}$ be two functions defined in the domain $\text{R}^+\cup\{0\}$. Find:
$\bigg(\frac{\text{f}}{\text{g}}\bigg)(\text{x})$
Answer
We have, $\text{f(x)}=\sqrt{\text{x}}$ and $\text {g(x)}=\text{x}$ be two functions defined in the domain $\text{R}^+\cup\{0\}.$
$\bigg(\frac{\text{f}}{\text{g}}\bigg)(\text{x})=\frac{\text{f(x)}}{\text{g(x)}}=\frac{\sqrt{\text{x}}}{\text{x}}=\frac{\text{1}}{\sqrt{\text{x}}}$
Let $\text{f(x)}=\sqrt{\text{x}}$ and $\text{g(x)}=\text{x}$ be two functions defined in the domain $\text{R}^+\cup\{0\}$. Find:
$\text{(f - g)}\text{(x)}$
Answer
We have, $\text{f(x)}=\sqrt{\text{x}}$ and $\text{g(x)}=\text{x}$ be two functions defined in the domain $\text{R}^+\cup\{0\}.$
$(\text{f}-\text{g})(\text{x})=\text{f(x)}-\text{g(x)}=\sqrt{\text{x}}-\text{x}$
Find the range of the following functions given by:
f(x) = 1 - x - 2
Answer
We know that,$|\text{x}-2|\geq0$
$\Rightarrow-|\text{x}-2|\leq0$
$\Rightarrow1-|\text{x}-2|\leq1$
$\Rightarrow\text{f(x)}\leq1$
$\therefore$ Range of f is $(-\infty, 11]$
Let $\text{f(x)}=\sqrt{\text{x}}$ and $\text{g(x)}=\text{x}$ be two functions defined in the domain $\text{R}^+\cup\big\{0\big\}$. Find:
$\text{(f+g)}\text{(x)}$
Answer
We have, $\text{f(x)}=\sqrt{\text{x}}$ and $\text{g(x)}=\text{x}$ be two functions defined in the domain $\text{R}^+\cup\big\{0\big\}.$
$(\text{f}+\text{g})(\text{x})=\text{f(x)}+\text{g(x)}=\sqrt{\text{x}}+\text{x}$
Given $\text{A} = \big\{1, 2, 3, 4, 5\big\}$$\text{S} = \text{(x, y)} : \text{x} \in\text{A}, \text{y} \in\text{A}.$ Find the ordered pairs which satisfy the condition given below: X + Y > 5
Answer
Given that: $\text{A} = \big\{1, 2, 3, 4, 5\big\}$ $\text{S} = \text{(x, y)} : \text{x} \in\text{A}, \text{y} \in\text{A}.$ X + Y > 5, So, the ordered pairs satisfying the given conditions are (4, 5), (5, 4), (5, 5).