Which function has the same minimum value as [tex]f(x)=\sqrt{x}+3[/tex]?
[tex]f(x)=3 x+3[/tex]
[tex]f(x)=|x|+3[/tex]
[tex]f(x)=\frac{1}{x}+3[/tex]
[tex]f(x)=-x^2+3[/tex]
Answer (1)
Determine the minimum value of f ( x ) = x + 3 , which is 3 at x = 0 .
Analyze the given functions to find their minimum values.
f ( x ) = ∣ x ∣ + 3 has a minimum value of 3 at x = 0 .
Therefore, the function with the same minimum value is f ( x ) = ∣ x ∣ + 3 .
Explanation
Problem Introduction We are given the function f ( x ) = x + 3 and we need to find which of the given functions has the same minimum value.
Finding Minimum Value of f(x) First, let's find the minimum value of f ( x ) = x + 3 . Since the square root function is only defined for non-negative values, the domain of f ( x ) is x ≥ 0 . The smallest value of x occurs when x = 0 , so the minimum value of f ( x ) is f ( 0 ) = 0 + 3 = 0 + 3 = 3 .
Analyzing Given Functions Now, let's analyze each of the given functions to find their minimum values:
f ( x ) = 3 x + 3 : This is a linear function. If there are no restrictions on x , it can take on values from − ∞ to ∞ . Therefore, it has no minimum value.
f ( x ) = ∣ x ∣ + 3 : The absolute value function ∣ x ∣ is always non-negative, and its minimum value is 0, which occurs at x = 0 . Therefore, the minimum value of f ( x ) = ∣ x ∣ + 3 is f ( 0 ) = ∣0∣ + 3 = 0 + 3 = 3 .
f ( x ) = x 1 + 3 : This function is not defined at x = 0 . As x approaches ∞ , f ( x ) approaches 3. As x approaches − ∞ , f ( x ) approaches 3. For 0"> x > 0 , 3"> f ( x ) > 3 , and for x < 0 , f ( x ) < 3 . Therefore, this function has no minimum value.
f ( x ) = − x 2 + 3 : This is a quadratic function with a negative leading coefficient, so it has a maximum value at its vertex. The vertex occurs at x = 0 , and the maximum value is f ( 0 ) = − 0 2 + 3 = 3 . As x goes to ± ∞ , f ( x ) goes to − ∞ , so there is no minimum value.
Comparison and Conclusion Comparing the minimum values, we see that f ( x ) = ∣ x ∣ + 3 has the same minimum value as the original function f ( x ) = x + 3 , which is 3.
Examples
Understanding minimum values of functions is crucial in optimization problems. For example, a company might want to minimize its production costs, which can be modeled as a function. By finding the minimum value of this cost function, the company can determine the most efficient way to operate. Similarly, in physics, finding the minimum potential energy of a system helps determine its stable equilibrium state. This concept is also used in machine learning to minimize the error of a model.
