Question

\(f(x)=\sqrt{x-3}+3\)

Short answer

The function \(f(x) = \sqrt{x - 3} + 3\) is based on the standard square root function, shifted three units to the right and translated three units upwards. Its domain and range are both [3, \(\infty\)), and it is concave up in this domain.

Step-by-step solution

Understanding the Function

The given function \(f(x) = \sqrt{x - 3} + 3\) is a combination of the square root function and a linear function offset. It can be considered as a transformation of the function \(g(x) = \sqrt{x}\). In detail, the function \(f(x)\) can be looked at as the function \(g(x)\) which is shifted three units to the right on the x-axis and then translated three units up on the y-axis.

Domain and Range of Function

The domain of the function \(f(x) = \sqrt{x - 3} + 3\) is the set of all x-values that makes the function defined. Since we cannot take the square root of a negative number in the real number system, the inside of the square root function, \((x - 3)\), has to be greater than or equal to zero, i.e., \(x - 3 \geq 0\). Solving for x, we get \(x \geq 3\). So the domain of the function is [3, \(\infty\)).\n\nFor the range of the function, the lowest value that \(f(x) = \sqrt{x - 3} + 3\) can have is when \(x = 3\), in which case \(f(x) = 3\). Since the square root function always yields non-negative values, it follows that as x gets progressively larger, \(f(x)\) also gets progressively larger since it is being incremented by the increasing square root component. Therefore, the range of the function is also [3, \(\infty\)).

Concavity of the Function

The function \(f(x) = \sqrt{x - 3} + 3\) can be seen as a square root function that has been translated. The basic square root function \(g(x) = \sqrt{x}\) is concave up in its domain. Therefore, the function \(f(x)\) which is derived from square root transformation is also concave up in its domain of [3, \(\infty\)).

Key concepts

Domain and Range of a Function

When studying functions, two foundational concepts to grasp are the domain and range. The domain encompasses all possible input values (x-values) for which the function is mathematically defined. In contrast, the range covers all potential output values (f(x) or y-values) the function can produce. For the function \(f(x)=\sqrt{x-3}+3\), we ascertain its domain by determining where the expression under the square root is non-negative, as square roots of negative numbers aren’t real values. Thus, we set \(x-3 \geq 0\), revealing that x must be greater than or equal to 3. The domain is therefore \([3, \infty)\).Meanwhile, the range is found by evaluating the smallest function value and observing its behavior as x increases. With square root functions, as x grows, so does its square root—resulting in larger f(x) values. The smallest value of f(x) occurs where x equals the lower bound of the domain, and since our function can only increase from there, the range mirrors the domain: \([3, \infty)\).Key Takeaway: The function \(f(x)\) defined above will accept any real number x greater than or equal to 3, and it will produce outputs starting at 3 and increasing indefinitely.

Square Root Graph Transformations

Graph transformations alter a function's visual representation without changing its core attributes. The function \(f(x)=\sqrt{x-3}+3\) emerges from transforming the parent square root function \(g(x)=\sqrt{x}\). This adaptation involves a horizontal shift to the right by 3 units and a vertical lift of 3 units. Why? Because subtracting from x within the square root shifts the graph right (opposite of what you might initially think!), while adding to the entire function moves it upward. Students should visualize graph transformations as sliding or stretching the curve of the original function. It helps to sketch the basic square root graph, then mentally (or on paper) shift it following these rules: right or left shifts correspond to adding or subtracting from x, and up or down shifts relate to adding or subtracting from the function itself.
Exercise Improvement Advice: A practical exercise could involve plotting the original square root function and incrementally applying these transformations, observing the resulting graph each step.

Concavity of Functions

Concavity refers to the curvature direction of a graph. A function is 'concave up' when its shape opens upwards like a bowl that can hold water, and 'concave down' when it opens downward. The square root function is a classic example of a function that is concave up everywhere in its domain. No matter how it's transformed—whether moving to the left, right, up, or down—it retains the concave up characteristic. For the function \(f(x)=\sqrt{x-3}+3\), even after shifting, it remains concave up through the domain \([3, \infty)\).This consistent concavity stems from the function's increasing rate of change; as x grows, the rate at which f(x) grows slows down. This results in a graph with a gradually flattening slope, which always curves upward. Recognizing concavity aids in understanding the behavior of the function over its domain and is particularly useful in areas like calculus, where it contributes to finding maxima and minima.
Tip: To quickly determine the concavity of a square root function, look for its increasing nature and remember that it will always curve up.