\frac{}{} is a LaTeX command, so needs to be inside [ tex] tags, like this:thereddevils said:Homework Statement
Given two functions f(x)=\frac{x}{x+1} , g(x)=\frac{x+2}{x}
thereddevils said:(1) find the domain of f and g
so D_f all real x , [tex]x\neq -1[/tex]
D_g is all real x , [tex]x\neq 0[/tex]
(2) Find the composite function gf and state its domain and range
g o f = 3+ 2/x , then its domain would be all x , [tex]x\neq 0[/tex] , am i correct ?
What is the easiest way to find its range , graph sketching ?
Mark44 said:\frac{}{} is a LaTeX command, so needs to be inside [ tex] tags, like this:
[tex]f(x)=\frac{x}{x+1}[/tex]
[tex]g(x)=\frac{x + 2}{x}[/tex]
Double-click either one to see what I did.
Part 1 is right, but part 2 isn't. (g o f)(x) = g(f(x)) = [tex]g(1 + 2/x)[/tex]
Think about what g does to an input...
Yes, what you have is right. I mistakenly put in the formula for g(x), but meant to put in the formula for f(x).thereddevils said:thanks Mark , but did u mean (g o f)(x)=g(x/(x+1)) ?
Right.thereddevils said:If so , i can see that g can take all real x , except for x=-1 from f and g itself has its own restriction so after combining , the domain of g o f would be all real x , except for x=-1 and x=0 .
thereddevils said:Am i correct ? so i can apply this way of checking for all such questions ?
Also , what's the easiest way to find the range for this question ? I can sketch the graph but would it be tedious to do so ?
Mark44 said:Yes, what you have is right. I mistakenly put in the formula for g(x), but meant to put in the formula for f(x).
Right.
[tex]g(f(x)) = g\left(\frac{x}{x + 1}\right) = 1 + \frac{\frac{x}{x+1} + 2}{\frac{x}{x+1}}[/tex]
The last expression can be simplified, but it's OK for our purposes right now. That expression is undefined if x = -1, because the two denominators in the rational expressions in the main numerator and denominator will be zero. The expression is also undefined if x = 0, since that would make the rational expression in the main denominator zero.
If you simplify that latter expression by multiplying top and bottom by (x + 1)/x over itself, you get f(g(x)) = 3 + 2/x, so it's no longer obvious that x can't be -1, but the only way that simplification could occur is if x is neither -1 nor 0.
Each of the graphs that make up the composite function is relatively easy to sketch. f(x) = x/(x + 1) = 1 - 1/(x + 1), by long division, and this is related to y = 1/x, but with a reflection and two translations, one horizontal and one vertical. Rf = {y | y != 1}.
g(x) = (x + 2)/x = 1 + 2/x, and this is also related to y = 1/x, with a stretch and a vertical translation. It turns out that Rg = {y | y != 1}.
The range of a function refers to all the possible output values, or y-values, that the function can produce for a given input. The domain of a function, on the other hand, refers to all the possible input values, or x-values, that the function can take. In other words, the range is the set of values that the function outputs, while the domain is the set of values that the function can take as input.
To find the range of a function, you can graph the function and look at the y-values that the function outputs. Alternatively, you can also plug in different input values and observe the corresponding output values. The domain of a function can be found by looking at the x-values that the function can take, which may be limited by the function's equation or restrictions on the input values.
Yes, the range of a function can include negative values. The range is simply the set of all possible output values, so it can include both positive and negative numbers, as well as zero.
Finding the range and domain of a function is important because it helps to identify the behavior and limitations of the function. It can also help in graphing the function and understanding its overall shape and characteristics. Additionally, knowing the range and domain can be useful in solving equations and finding the inverse of a function.
In some cases, the range and domain of a function can be the same. For example, a linear function with a slope of 1 will have a range and domain that are both equal to all real numbers. However, this is not always the case as some functions may have a restricted domain or a limited range due to their equations or other factors.