Answer:
Domain : (- ∞, - 5), and (3 / 2, ∞),
Range : (∞, 0.4437]
Step-by-step explanation:
Assuming that we want our answer in interval notation, let's start by determining the domain. Remember that the domain can be found where the function is undefined.
Given : f(x) = (2x - 7) / (2x² + 7x - 15)
Alternative Form : (2x - 7) / (x + 5)(2x - 3)
To receive this 'alternative form' we can simply factor the expression 2x² + 7x - 15. See the procedure below,
![\mathrm{Given : 2x^2+7x-15} ,](https://tex.z-dn.net/?f=%5Cmathrm%7BGiven%20%3A%202x%5E2%2B7x-15%7D%20%2C)
![\mathrm{Break\:the\:expression\:into\:groups : \left(2x^2-3x\right)+\left(10x-15\right)} ,](https://tex.z-dn.net/?f=%5Cmathrm%7BBreak%5C%3Athe%5C%3Aexpression%5C%3Ainto%5C%3Agroups%20%3A%20%5Cleft%282x%5E2-3x%5Cright%29%2B%5Cleft%2810x-15%5Cright%29%7D%20%2C)
![\mathrm{Factor\:out\:}x\mathrm{\:from\:}2x^2-3x\mathrm{:\quad }x\left(2x-3\right),\\\mathrm{Factor\:out\:}5\mathrm{\:from\:}10x-15\mathrm{:\quad }5\left(2x-3\right),](https://tex.z-dn.net/?f=%5Cmathrm%7BFactor%5C%3Aout%5C%3A%7Dx%5Cmathrm%7B%5C%3Afrom%5C%3A%7D2x%5E2-3x%5Cmathrm%7B%3A%5Cquad%20%7Dx%5Cleft%282x-3%5Cright%29%2C%5C%5C%5Cmathrm%7BFactor%5C%3Aout%5C%3A%7D5%5Cmathrm%7B%5C%3Afrom%5C%3A%7D10x-15%5Cmathrm%7B%3A%5Cquad%20%7D5%5Cleft%282x-3%5Cright%29%2C)
![x\left(2x-3\right)+5\left(2x-3\right) = \left(2x-3\right)\left(x+5\right) - \mathrm{Factored\:expression}](https://tex.z-dn.net/?f=x%5Cleft%282x-3%5Cright%29%2B5%5Cleft%282x-3%5Cright%29%20%3D%20%5Cleft%282x-3%5Cright%29%5Cleft%28x%2B5%5Cright%29%20-%20%5Cmathrm%7BFactored%5C%3Aexpression%7D)
Now let's find the domain using the expression '(x + 5)(2x - 3) = 0.' If the denominator equals 0, the function is considered undefined.
![\mathrm{Given : \left(x+5\right)\left(2x-3\right)=0} ,\\x+5=0:\quad x=-5, 2x-3=0:\quad x=\frac{3}{2}\\\mathrm{The\:solutions\:are: x=-5,\:x=\frac{3}{2}}](https://tex.z-dn.net/?f=%5Cmathrm%7BGiven%20%3A%20%5Cleft%28x%2B5%5Cright%29%5Cleft%282x-3%5Cright%29%3D0%7D%20%2C%5C%5Cx%2B5%3D0%3A%5Cquad%20x%3D-5%2C%202x-3%3D0%3A%5Cquad%20x%3D%5Cfrac%7B3%7D%7B2%7D%5C%5C%5Cmathrm%7BThe%5C%3Asolutions%5C%3Aare%3A%20x%3D-5%2C%5C%3Ax%3D%5Cfrac%7B3%7D%7B2%7D%7D)
Knowing these solutions the domain has the intervals (- ∞, - 5), and (3 / 2, ∞). The range is the set of values that correspond to the domain, so in this case the range would be (∞, 42 + 8√17 / 169]. 42 + 8√17 / 169 = (About) 0.4437, so it lies on the interval (∞, 0.4437].
The images below represent the plotted function in two areas. There are 3 curves in this graph.