# Comparing f(x) and g(x) for All x in R

• Dank2
In summary, the functions f(x) = 5(√(x^2 +1)) and g(x) = 3x + 4 have a minimum point at x=3/4, with f(x) being greater than or equal to g(x) for all real values of x. This can be shown by finding the minimum of the difference function f(x) - g(x) and observing that it is equal to 0 at x=3/4, making f(x) greater than or equal to g(x).
Dank2

## Homework Statement

the two functions $$f(x) = 5(√(x^2 +1))$$ $$g(x) = 3x + 4$$.

## The Attempt at a Solution

I can get the minimum point of f(x) and it is bigger than g(x) and that point, however g(x) is tangential to the curve f(x) at point 3/4.
what else do i miss to show that f(x) is bigger or equal than g(x) for all x in R?

Dank2 said:

## Homework Statement

the two functions $$f(x) = 5(√(x^2 +1))$$ $$g(x) = 3x + 4$$.

## The Attempt at a Solution

I can get the minimum point of f(x) and it is bigger than g(x) and that point, however g(x) is tangential to the curve f(x) at point 3/4.
what else do i miss to show that f(x) is bigger or equal than g(x) for all x in R?

Find the minimum of the difference function ##f(x) - g(x) = 5 \sqrt{x^2+1} -(3x+4)##.

BTW: the word is parabola, not parabula; and anyway, you do not have a parabola anywhere in this problem.

Dank2
Ray Vickson said:
Find the minimum of the difference function ##f(x) - g(x) = 5 \sqrt{x^2+1} -(3x+4)##.

BTW: the word is parabola, not parabula; and anyway, you do not have a parabola anywhere in this problem.
thanks

its point 3/4. and it is the absolute minimum of the graph that's equal to 0, therefore f(x) >= g(x).

Dank2 said:
thanks

its point 3/4. and it is the absolute minimum of the graph that's equal to 0, therefore f(x) >= g(x).
Right!

## 1. How do I compare f(x) and g(x) for all values of x in the set of real numbers?

To compare f(x) and g(x) for all values of x in the set of real numbers, you can graph both functions on the same coordinate plane and observe their behavior. Alternatively, you can analyze their algebraic expressions and determine their similarities and differences.

## 2. What are the key differences between f(x) and g(x) for all values of x in the set of real numbers?

The key differences between f(x) and g(x) for all values of x in the set of real numbers depend on the specific functions. Generally, these differences can be observed through their graphs, such as differences in shape, intercepts, and slopes.

## 3. How can I determine which function, f(x) or g(x), is larger for a specific value of x in the set of real numbers?

To determine which function, f(x) or g(x), is larger for a specific value of x in the set of real numbers, you can plug in that value for x and compare the resulting values. The greater value corresponds to the larger function.

## 4. Can I compare f(x) and g(x) for all values of x in the set of real numbers if they have different domains?

It is possible to compare f(x) and g(x) for all values of x in the set of real numbers even if they have different domains. This can be done by finding the common domain between the two functions and comparing them within that shared interval.

## 5. What is the purpose of comparing f(x) and g(x) for all values of x in the set of real numbers?

Comparing f(x) and g(x) for all values of x in the set of real numbers allows us to analyze and understand the behavior and relationship between the two functions. It can also help us determine which function is more appropriate for a given scenario or problem.

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