Another polynomial function

In summary, the function f(x) = √(x+2) - √(x-1) is defined for all real numbers greater than or equal to 1 and has a range from 0 to √3. Through various mathematical techniques such as calculating specific values, taking the derivative, and considering limits, it can be shown that the function has a positive derivative, a limit of 0 as x approaches infinity, and a limit of 1 as x approaches 1. This information can help us better understand
  • #1
mtayab1994
584
0

Homework Statement



f:[1,+∞[→ℝ
x→sqrt(x+2)-sqrt(x-1)

Homework Equations



show that f([1,+∞[)=]0,sqrt(3)]

The Attempt at a Solution



Any tips on how to start it.
 
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  • #2
mtayab1994 said:

Homework Statement



f:[1,+∞[→ℝ
x→sqrt(x+2)-sqrt(x-1)


Homework Equations



show that f([1,+∞[)=]0,sqrt(3))


The Attempt at a Solution



Any tips on how to start it.

What have you done so far?

RGV
 
  • #3
Ray Vickson said:
What have you done so far?

RGV
well nothing really, i want something that'll help me get going.
 
  • #4
things to try:

1. calculate f(x) explicitly for a few values of x. i always like 0,1,-1 and 42 (ok, 0,1 and-1 won't work. pick something else. maybe 2,4 and 6)

2. see if lim x→∞ and lim x→1+ exist.

3. see if f(x) has a global maximum or minimum (yeah, derivatives, we can use them, right?)
 
  • #5
Deveno said:
things to try:

1. calculate f(x) explicitly for a few values of x. i always like 0,1,-1 and 42 (ok, 0,1 and-1 won't work. pick something else. maybe 2,4 and 6)

2. see if lim x→∞ and lim x→1+ exist.

3. see if f(x) has a global maximum or minimum (yeah, derivatives, we can use them, right?)

alright so these 3 steps should help me?
 
  • #6
you want to figure out how f(x) behaves. you need to get some information.
 
  • #7
yea but in class when we want to solve something like this he have to show that f(x)=y or f(x)≥0.
 
  • #8
what is f(1)?

we need to establish "some" things about f. it's going to be impossible to do, if all you think is "well, f is a function".

take it's derivative. it is always positive, does it ever equal 0, is it some places positive, and some places negative? what does that tell you about f?

are there any places where f(x) = 0 (does this have anything to do with whether or not f crosses the x-axis)?

what happens when x gets "really really big"? does it have a limit as x→∞? if so, what is this limit?
 

1. What is a polynomial function?

A polynomial function is a mathematical function that is defined by a finite sum of terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power. In simpler terms, it is an algebraic expression that contains variables and constants, and the operations of addition, subtraction, and multiplication.

2. How do you graph a polynomial function?

To graph a polynomial function, you will need to plot points on a coordinate plane. The number of points you need to plot will depend on the degree of the polynomial. You can also use the leading coefficient, degree, and intercepts to help you graph the function accurately.

3. What is the degree of a polynomial function?

The degree of a polynomial function is the highest exponent or power of the variable in the expression. It is also the number of terms in the polynomial.

4. Can a polynomial function have negative exponents?

No, a polynomial function cannot have negative exponents. This is because a polynomial function is defined as having non-negative integer powers on its variables. If it has negative exponents, it would become a rational function instead.

5. What are the different types of polynomial functions?

The two main types of polynomial functions are monomials and binomials. Monomials have one term, while binomials have two terms. Other types of polynomial functions include trinomials (three terms), quadrinomials (four terms), and so on. They can also be classified based on their degree, such as linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on.

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