find the domain of the function?

Question

g(x)=ln (2+x-x^2)

Answer 1

x^2-x-2=0
(x-2)(x+1)=0
So the zero's occur at x=2 and x=-1
Therefore, the domain is: -1

Answer 2

g(x) = ln(2 + x - x^2)

To find the domain of this function, it is required that the argument of the log is strictly greater than 0. That is,

2 + x - x^2 > 0

Solve this inequality.

-x^2 + x + 2 > 0
-(x^2 - x - 2) > 0
-(x - 2)(x + 1) > 0

Multiply both sides by (-1) flips the inequality.

(x - 2)(x + 1) < 0

To solve this inequality, we need to find the critical numbers; critical numbers make the left hand side equal to 0, and they are -1 and 2. Make a number line consisting of these values.

. . . . . . . . (-1) . . . . . . . . . . . .(2) . . . . . . . . . . .

Test each region to determine the sign of (x - 2)(x + 1).
Test x = -2: (-2 - 2)(-2 + 1) = (-4)(-1) = 4, which is positive.
Mark the region as positive.

. . . {+} . . . . (-1) . . . . . . . . . . . .(2) . . . . . . . . . . .

Test x = 0: (0 - 2)(0 + 1) = (-2)(1) = -2, which is negative. Mark the region as negative.

. . . {+} . . . . (-1) . . . . .{-} . . . . . .(2) . . . . . . . . . . .

Test x = 3: (3 - 2)(3 + 1) = [positive].

. . . {+} . . . . (-1) . . . . .{-} . . . . . .(2) . . . . .{+}. . . . .

Our inequality is asking when (x - 2)(x + 1) < 0; i.e. it's asking when it is negative. Look at the negative regions on the number line; it's the region between -1 and 2.

Therefore, the domain of the function is the set of all x such that -1 < x < 2. In interval notation, (-1, 2).

Answer 3

we just require 2 + x - x^2 to be positive. This means we want 2 + x to be larger than x^{2}. Use quadratics.

2 + x - x^{2} > 0.
so x^{2} - x - 2 < 0.
(x + 1)(x -2) < 0.

If x > 2 everything on the left is positive and so it's no good. If x < -1 then both terms on the left are negative with a positive product, that's no good. We also have to exclude the left hand side EQUALING zero, i.e. when x = 2 or x = -1. So we end up with -1
to check: pick an x between -1 and 2 and check that an answer is defined on your calc. Then try x = -1 and x = 2. These shouldn't work. Also try x < -1 and try x > 2. Good Luck.

Answer 4

-1

Answer 5

graph the function and see where the x doesn't go