Understanding the Domain and Range of the Function f(x) √(4 - 2x) 5

In this article, we explore the domain and range of the mathematical function f(x) sqrt{4 - 2x} 5, providing a detailed analysis for SEO optimization.

Introduction to the Function

The function in question is f(x) sqrt{4 - 2x} 5. This expression involves a square root and an addition of a constant, which requires careful analysis to determine its valid domain and range.

Understanding the Domain

The domain of a function is the set of all input values (x-values) for which the function is defined. In the case of this function, the primary concern is the square root term, sqrt{4 - 2x}. The expression inside the square root, 4 - 2x, must be non-negative for the square root to be defined in the real number system.

Mathematically, this can be represented as:

4 - 2x ≥ 0

Solving this inequality for x:

begin{align*} 4 ≥ 2x 2 ≥ x x ≤ 2 end{align*}

Therefore, the domain of the function is:

(-infty

The domain of the function, highlighting the interval (-infty

Exploring the Range

The range of a function is the set of all possible output values (y-values) that the function can produce. For the function f(x) sqrt{4 - 2x} 5, we need to find the minimum and maximum values of the output as x varies over its domain.

The Minimum Value of the Function

The minimum value of the square root term, sqrt{4 - 2x}, occurs when the expression inside the square root is zero. This happens when:

4 - 2x 0 x 2

Substituting x 2 into the function, we get:

f(2) sqrt{4 - 2(2)} 5 sqrt{0} 5 0 5 5

Therefore, the minimum value of the function is 5.

The Maximum Value of the Function

As x approaches negative infinity, the expression 4 - 2x increases without bound because -2x becomes a very large positive number. Consequently, the square root term sqrt{4 - 2x} also increases without bound.

Mathematically, as x rightarrow -infty, f(x) rightarrow infty. Therefore, there is no upper bound for the function’s values.

The range of the function is thus:

[5, infty]

The range of the function, showing the interval [5, infty].

Conclusion

In summary, we have determined that the domain of the function f(x) sqrt{4 - 2x} 5 is (-infty

For further exploration, consider studying more complex functions involving square roots, logarithms, and trigonometric functions. Each mathematical function offers unique insights into different domains of analysis.