You know, when we first dive into the world of functions in math, it can feel a bit like trying to understand a secret language. We see these equations, like g(x) = 7 / (6 - 5x), and we're asked to find its "domain." What does that even mean?
Think of a function as a little machine. You put something in (an input, usually represented by 'x'), and it gives you something out (an output, often 'y' or 'g(x)'). The domain is simply the collection of all the possible things you can safely put into that machine without breaking it.
So, what could break our g(x) machine? Well, the biggest culprit in many functions, especially those with fractions, is division by zero. Nobody likes that, and mathematically, it's undefined. Looking at g(x) = 7 / (6 - 5x), the problem lies in the denominator: 6 - 5x. If this part becomes zero, our function throws a fit.
To figure out when this happens, we just set the denominator equal to zero and solve for 'x':
6 - 5x = 0
If we move things around a bit, we get 5x = 6, and then x = 6/5.
This tells us that if we try to plug 6/5 into our function, the denominator will be zero, and the function won't be defined. It's like trying to put a square peg in a round hole – it just doesn't work.
Therefore, the domain of g(x) is all real numbers except for 6/5. We can write this in a couple of ways. Using set notation, it looks like {x | x ∈ ℝ, x ≠ 6/5}, which reads as "the set of all x such that x is a real number and x is not equal to 6/5." Or, if you prefer intervals, it's (-∞, 6/5) ∪ (6/5, +∞). This means we can use any number from negative infinity up to (but not including) 6/5, and then any number from (but not including) 6/5 all the way to positive infinity.
It's a bit like having a road with a single pothole. You can drive on almost the entire road, but you have to steer clear of that one specific spot. That's the essence of finding a function's domain – identifying all the valid inputs that keep the function running smoothly.
