Find domain and range
f(x){-1,x<=1
3x+2,-1<x<1
7-2x, x>=1}
Can someone please help me with this, I'm not sure how to go about solving this problem.
Hello,
For the domain, it's just the values x can take.
For the range... if x<=-1, then f(x)=-1. Thus the range contains -1.
f(x)=3x+2 if -1<x<1, that is to say -1<3x+2<5. This is part of the range too.
f(x)=7-2x if x>=1, that is to say 7-2x<=5. This actually gives the whole range.
Your function is :
$\displaystyle f(x)=\left\{ \begin{array}{lll} -1 \quad & \quad x \le -1 \\ 3x+2 \quad & -1<x<1 \\ 7-2x \quad & \quad x \ge 1 \end{array} \right.$
$\displaystyle x \le -1$ is like $\displaystyle ]-\infty~,-1]$, $\displaystyle -1<x<1$ is like $\displaystyle (-1~,1)$ and $\displaystyle x \ge 1$ is like $\displaystyle [1~,+\infty[$
So x can be any number in $\displaystyle \mathbb{R}$ (the set of real numbers).
Why would the range have been only -1 and 1 ? On the contrary, it's the values that would have been controversary because we're not sure the function is continuous at these points. In fact, it is.
You're told that if x>=1, then f(x)=7-2x. So find out what values 7-2x can take if x>=1. This gives 7-2x<=5
Okay, maybe I should start from the beginning, for the given function, isn't the first number (or expression) the lowest number that can be used for the domain? And the second, the largest? That's why I picked -1 and 1 because in the first expression the number is -1 and in the last it is 1. I'm just really not sure. Isn't the range the result of what you put into the function?
Yes. This means that you want the different values that f(x) can have.
For any value of x that is inferior to -1, the function is -1
For any value of x that is superior to -1 and inferior to 1, we have :
$\displaystyle -1<x<1 \implies -3<3x<3 \implies -1<\underbrace{3x+2}_{f(x) \text{ for } -1<x<1}<5$. So the different values of f(x) while -1<x<1 are all the values within -1 and 5.
So far, the range is $\displaystyle [-1~,5)$.
For any value of x that is superior to 1, we have :
$\displaystyle x \ge 1 \implies 2x \ge 2 \implies -2x \le -2 \implies \underbrace{7-2x}_{f(x) \text{ for } x \ge 1} \le 5$. So the different values of f(x) while $\displaystyle x \ge 1$ are all the values inferior to 5.
So we have $\displaystyle f(x) \le 5$ if $\displaystyle x \ge 1$, which $\displaystyle f(x) \in (-\infty~,5]$
The range will be the union of these two intervals :
$\displaystyle R=[-1~,5) \cup (-\infty~,5]=(-\infty~,5]$
Is it clear enough ?