Let f(x)=(3x-7)/(x+1)
What is the range of f?
Answer in interval notation.
(The domain is (-inf,-1) U (-1,inf)
You stated the domain, it asked for the range
range:
y < 3 OR y > 3 , y ∈ R
or
range is {y| y ∈ R, y ≠3 }
I will let you change that to interval notation, not really familiar with that notation. We did not have that in the "olden days" when I was teaching.
To find the range of the function f(x) = (3x - 7)/(x + 1), we need to determine the set of all possible values that the function can output.
The first step is to find any restrictions on the function. In this case, the function has a restriction where the denominator (x + 1) cannot be zero since division by zero is undefined. So, we should exclude x = -1 from the domain.
Now, let's find the range of f(x). We can start by considering what happens as x approaches positive infinity and negative infinity.
As x approaches positive infinity, the value of (3x - 7)/(x + 1) also approaches positive infinity. This means that the function has no upper bound and the range extends to positive infinity.
As x approaches negative infinity, the value of (3x - 7)/(x + 1) also approaches negative infinity. This means that the function has no lower bound and the range extends to negative infinity.
Therefore, the range of the function f(x) is from negative infinity to positive infinity, which can be expressed in interval notation as (-∞, ∞).
To determine the range of the function f(x) = (3x - 7)/(x + 1), we need to analyze the possible output values, also known as the range, based on the given domain.
Start by considering the behavior of the function as x approaches positive infinity (+∞) and negative infinity (-∞). Take the limit of f(x) as x approaches both infinities:
1. As x approaches +∞:
lim(x→+∞) (3x - 7)/(x + 1)
To evaluate this limit, divide both the numerator and denominator by the highest power of x, which is x in this case:
= lim(x→+∞) (3 - 7/x)/(1 + 1/x)
Now, as x approaches infinity, both (7/x) and (1/x) approach zero, so the limit simplifies to:
= lim(x→+∞) (3 - 0)/(1 + 0)
= 3/1
= 3
So, as x approaches positive infinity, f(x) approaches 3.
2. Similarly, as x approaches -∞:
lim(x→-∞) (3x - 7)/(x + 1)
= lim(x→-∞) (3 - 7/x)/(1 + 1/x)
= (3 - 0)/(1 + 0)
= 3/1
= 3
Thus, as x approaches negative infinity, f(x) also approaches 3.
From the above analysis, we observe that the range of f(x) is the set of all possible output values, which is:
Range of f: {3}
In interval notation, we can represent this range as:
Range of f: (3, 3)
Please note that the range is a single value (3) since the function is continuous and does not have any vertical asymptotes or holes in the graph.