Answer:
The minimum of h(x) is farther left and down from the minimums of f(x) and g(x) ⇒ answer 4
Explanation:
* Lets revise the meaning of the minimum of the function
- The minimum of a function is the lowest point of a vertex
- The minimum value is the y-coordinate of the lowest point
* Now lets solve the problem
∵ f(x) = √x
- The domain of the function is all the real numbers ≥ 0, because
there is no square root for negative numbers
- We will use the first value of x to find the range of the function
∵ x = 0 ⇒ first value of the domain
∴ f(0) = √0 = 0
∴ The minimum of f(x) is point (0 , 0)
∵ g(x) = √(x - 3) + 1
- Lets find the domain of the function
∵ x - 3 ≥ 0 ⇒ add 3 for both sides
∴ x ≥ 3
∴ The domain of the function is all real values of x ≥ 3
- We will use the first value of x to find the range of the function
∵ x = 3 ⇒ the first value of the domain
∴ g(3) = √(3 - 3) + 1 = 0 + 1 = 1
∴ The minimum of g(x) is point (3 , 1)
∵ h(x) = √(x + 1) - 2
- Lets find the domain of the function
∵ x + 1 ≥ 0 ⇒ subtract 1 for both sides
∴ x ≥ -1
∴ The domain of the function is all real values of x ≥ -1
- We will use the first value of x to find the range of the function
∵ x = -1 ⇒ the first value of the domain
∴ h(-1) = √(-1 + 1) - 2 = 0 - = -2
∴ The minimum of h(x) is point (-1 , -2)
* To find the correct answer look to the minimum points of the
functions (0 , 0) , (3 , 1) , (-1 , -2)
- The minimum of f(x) lies on the origin
- The minimum of g(x) lies in the 1st quadrant
- The minimum of h(x) lies in the 3rd quadrant
∴ The minimum of h(x) is farther left and down from the minimums
of f(x) and g(x)
- Look to the attached graph to more understand
- The red curve is f(x)
- The blue curve is g(x)
- The green curve is h(x)