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1-40.

State the domain for each of the functions below.

1. $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$

We all know that $0$ is excluded from the denominator of a fraction. What value of $x$, if any, would make the denominator $x^2 + 1 = 0$?

1. $g ( x ) = \frac { 1 } { x } - \frac { x } { x + 1 }$

Consider the domain of each part of the subtraction problem separately.
Since $g(x)$ is the difference between $\frac{1}{x}$ and $\frac{x}{x+1}$, both excluded values must be excluded from $g(x)$.

1. $h ( x ) = \sqrt { x ^ { 2 } - 9 }$

Typically, a square root function has a domain of $x ≥ 0$, but in this case, $x$ is squared... meaning both positive or negative values will yield positive outputs. However, do not neglect to consider the $-9$.
What values of $x$, both positive and negative, will make $x^2 - 9 > 0$?

$x ≤ − 3\text{ U } x ≥ 3$

1. $k ( x ) = \frac { \operatorname { log } ( x - 3 ) } { \sqrt { x + 4 } }$

Consider the domain of the numerator and the domain of the denominator separately.
I. What is the domain of $\operatorname{log}(x - 3)$?
III. What value is excluded from the denominator of all fractions?
IV. Now combine these results to find the domain of $k(x)$.

I. $x > 3$
II. $x ≥ -4$
III. $x ≠ -4$
IV. $x > 3$