Another question about domains and range
how would I find the range of (3x-4)/(6x-1) ? Without using a graph
the domain is x can't be 1/6
What is the range?
without using a graph
The range is how high or how low your graph can go
So look at what happens when you use large values of x
e.g. in your mind try x = 100 000
so you would have 299 996/599 999
For all practical purposes this is 3/6 or 1/2, even though you will never reach it.
As a matter of fact try it on your calculator you will get .49999... , in other words, less than 1/2
The larger the value of x, the closer we will get to 1/2
If we try large negative values of x
say, x = -100 000
then our calculation is -300 006/-600 001
which is slightly over .5
so no value of x will produce a y equal to 1/2
so the domain is the set of all real values of y ,except y = .5
(if you try to solve
(3x-4)/(6x-1) = 1/2
we get
6x - 8 = 6x-1
-8 = -1 which is false and a contradiction.
So there is no solution)
1. Use a calculator to approximate the square root of 320. Round to three decimal places
To find the range of a function, we need to determine all the possible output values or y-values that the function can produce. In this case, the function is given by (3x - 4)/(6x - 1).
To find the range, we can start by analyzing the behavior of the function as x approaches positive or negative infinity. Let's examine the numerator and denominator separately:
As x approaches positive infinity, the term 3x dominates the -4 term in the numerator. Hence, the numerator becomes infinitely large, and likewise, the denominator (6x - 1) also becomes infinitely large. Therefore, the function approaches positive infinity as x approaches positive infinity.
Similarly, as x approaches negative infinity, the term 3x dominates the -4 term in the numerator. In this case, the numerator becomes infinitely small (negative infinity), and the denominator (6x - 1) also becomes infinitely large (negative infinity). Thus, the function approaches negative infinity as x approaches negative infinity.
From these observations, we can conclude that the range of the function is the set of all real numbers, except for infinity. In other words, the range is (-∞, +∞).
Note that the range of the function does not include any particular restriction or exclusions, except for the infinite values. Therefore, we can say that the range is all real numbers except infinity.