how would i go about finding the domain of the function
f(x)=(3x+1)/(sqrt(x^2+x-2))
and both the domain and range of
g(x)=(5x-3)/(2x+1)
Well, the domain would be all x except where the denominator is zero or the sqrt is of a negative number so we need to find where the denominator is zero and where x^2 + x - 2 is negative.
let's look at that denominator function inside the radical sign:
d = x^2+x -2
It is a parabola that opens up
first find the zeros
0 = (x-1)(x+2)
so it is zero at x = -2 and x =+1
those points must not be in the domain but also all the points between them are out because the d function dips below zero between -2 and +1
so our domain is -2 > x > +1
-----------------------------
Now for the second one the denominator is zero for x = -1/2 and all real x except x = -1/2 is the domain
To find the range, sketch the function
for x << 0, g(x)--> +5/2
for x >> 0, g(x)--> +5/2
for x = 0, g(x) = -3
now look at where the numerator is zero
like
x = .6, g = 0
now look at points close to x = -.5 where g(x) gets huge
like
x = -.6, g = 30
x = -.4, g = -25
that tells you what happens each side of the singularity at x = -.5
We see that g(x) goes to +oo as x approaches -.5 from the left and g(x) goes to -oo as x approaches -.5 from the right.
Therefore the range of g is from -oo to +oo
i understood the whole first question and then i understood the domain of the second question. i did get confused though when it came to finding the range because the teacher gave us the answers w/o explanations and it said that the range was all numbers except for 5/2. i'm not sure how she got that.
thanks for all your help!
To find the domain of a function, you need to identify any values of x that would result in an undefined expression. In the case of f(x) = (3x +1) / √(x^2 + x - 2), the expression is undefined if the denominator (√(x^2 + x - 2)) is equal to zero.
To find when the denominator is equal to zero, you solve the equation x^2 + x - 2 = 0. Factoring this quadratic equation, you get (x - 1)(x + 2) = 0. So, x = 1 or x = -2.
Therefore, the domain of f(x) is all real numbers except x = 1 and x = -2.
Now, moving to the function g(x) = (5x - 3) / (2x + 1), the domain is the set of all real numbers excluding any values that would make the denominator equal to zero (to avoid division by zero).
To find these values, we solve the equation 2x + 1 = 0. Subtracting 1 from both sides gives 2x = -1, and then dividing by 2 gives x = -1/2.
So, the domain of g(x) is all real numbers except x = -1/2.
To find the range of a function, you need to analyze the behavior as x approaches positive infinity and negative infinity. However, in the case of g(x) = (5x - 3) / (2x + 1), since both the numerator and denominator have degree 1, the range will be the set of all real numbers.
Therefore, the range of g(x) is all real numbers.