Problem 51 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. Then sketch its graph and find the range. $$y=5 \sqrt{x}-2$$

Step-by-Step Solution

Answer

The domain of the function \(y=5 \sqrt{x}-2\) is \([0, \infty)\) and the range is \([-2, \infty)\). The graph of the function starts from (0,-2) and increases slowly as \(x\) increases.

Step 1: Determine the Domain

Since the function has a square root, and the square root of a negative number is undefined in the real number system, to get the domain, set \(x \geq 0\). So the domain is all real numbers greater than or equal to zero.

Step 2: Sketch the graph

Begin at the point (0,-2) since when \(x = 0\), \(y = 5 \sqrt{0} - 2 = -2\). The graph should start from this point and it rises slowly since the function involves a square root.

Step 3: Determine the Range

Looking at the graph, the values of \(y\) will always be greater than or equal to -2 because 5 times the square root of a non-negative number \(x\), minus 2 will always be -2 or more. Hence, the range of the function is \([-2, \infty)\).

Key Concepts

Square Root Function

The square root function is one of the fundamental functions in mathematics. It is denoted by the square root symbol \( \sqrt{} \). To understand this function better, think of it as asking the question: "What number multiplied by itself gives this value?". For instance, the square root of 9 is 3, because \(3 \times 3 = 9\).

In our specific function, \(y = 5 \sqrt{x} - 2\), the square root function is transformed by multiplying it by 5 and then subtracting 2 from the result. This transformation affects the graph's steepness and its starting point on the plane. Multiplying by 5 means the graph rises five times faster than \(y = \sqrt{x}\) alone, while subtracting 2 shifts the entire graph downward by 2 units.

Graph Sketching

Graph sketching is a useful skill that helps you understand the behavior of a function intuitively. When sketching the graph of \(y = 5 \sqrt{x} - 2\), start at the point where \(x = 0\).

- At \(x = 0\), the function simplifies to \(y = 5 \times 0 - 2 = -2\). Thus, your graph begins at point \((0, -2)\) on the coordinate plane.- As \(x\) increases, the value of \(y\) becomes greater due to the square root component. However, since square roots grow more slowly compared to linear relationships, your graph will gradually rise, approaching infinity as \(x\) increases.
This sketching process helps you visualize the gradual increase and understand how different parts of the equation, such as the multiplier (5) and constant (-2), affect the graph's shape.

Function Domain

The domain of a function tells us what values \(x\) can take. For the function \(y = 5 \sqrt{x} - 2\), there's a critical rule: we cannot take the square root of a negative number in the real number system.

Therefore, \(x\) must be zero or positive, because \(\sqrt{x}\) is only defined when \(x \geq 0\).

• The domain of this function is all real numbers starting from zero, moving positively towards infinity.

You can write the domain in interval notation as \([0, \infty)\). Considering the function's definition helps to avoid calculation errors and better understand where the function graph can be drawn.

Function Range

The range of a function defines what values \(y\) can take as a result of substituting different \(x\) into the function. For the function \(y = 5 \sqrt{x} - 2\), it’s crucial to see how the function behaves to determine its range.

- With our starting point at \((0, -2)\), the smallest value \(y\) can be is -2, when \(x = 0\).- As \(x\) increases, \(y\) will grow larger, since both the square root function and positive multiplier (5) lead to increasing values.

Therefore, the range of the function \(y = 5 \sqrt{x} - 2\) is all real numbers that are greater than or equal to -2. Expressed in interval notation, the range is \([-2, \infty)\). Recognizing how transformations in the function equation affect the range is key to mastering graph sketching and functional analysis.