Question

What is the domain (in interval notation) of the following functions?

1. g(x)=3/(5x-4)
2. h(x)=√(x)/(x-5)
3. f(x)=√(x)/(x^2-5x)
4. g(x)=(√(x)+5)/(x^2-x-20)
5. h(x)=3/(x^2+1)
6. f(x)=(√(x-2))/(x+1)
7. g(x)= x^2/(3x^2-x-2
8. h(x)=3(x-4)^2-7
Number sets in parenthesis are either on top of or beneath the fraction bar and ^2 here represents a number squared.

Answer 1

`1.\\g(x)=(3)/(5x-4)\\\\D:5x-4\\eq0\to5x\\eq4\ \ \ /:5\to x\\eq(4)/(5)\to x\in\mathbb{R}\ \backslash\ \{(4)/(5)\}\\\\2.\\h(x)=(√(x))/(x-5)\\\\D:x\geq0\ \wedge\ x-5\\eq0\to x\geq0\ \wedge\ x\\eq5\to x\in\left<0;\ \infty\right)\ \backslash\ \{5\}`


`3.\\f(x)=(√(x))/(x^2-5x)\\\\D:x\geq0\ \wedge\ x^2-5x\\eq0\to x\geq0\ \wedge\ x(x-5)\\eq0\\\\\to x\geq0\ \wedge\ x\\eq0\ \wedge\ x\\eq5\to x\in\mathbb{R^+}\ \backslash\ \{5\}\\\\4.\\g(x)=(√(x)+5)/(x^2-x-20)\\\\D:x\geq0\ \wedge\ x^2-x-20\\eq0\to x\geq0\ \wedge\ (x+4)(x-5)\\eq0\\\\\to x\geq0\ \wedge\ x\\eq-4\ \wedge\ x\\eq5\to x\in\left<0;\ \infty\right)\ \backslash\ \{-4;\ 5\}`


`5.\\h(x)=(3)/(x^2+1)\\\\D:x^2+1\\eq0\to x^2\\eq-1\to x\in\mathbb{R}\\\\6.\\f(x)=(√(x-2))/(x+1)\\\\D:x-2\geq0\ \wedge\ x+1\\eq0\to x\geq2\ \wedge\ x\\eq-1\to x\in\left<2;\ \infty\right)`


`7.\\g(x)=(x^2)/(3x^2-x-2)\\\\D:3x^2-x-2\\eq0\to (3x+2)(x-1)\\eq0\to x\\eq-(2)/(3)\ \wedge\ x\\eq1\\\\\to x\in\mathbb{R}\ \backslash\ \{-(2)/(3);\ 1\}\\\\8.\\h(x)=3(x-4)^2-7\\\\D:x\in\mathbb{R}`