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Chapter 1: Problem 26

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y=\sqrt{2 x-1} $$

Short Answer

Expert verified
The x-intercept of the function is \(x = 0.5\) and there is no y-intercept for this function.

Step by step solution

01

Find the x-intercept

Setting \(y = 0\) in our function, we get: \[0 = \sqrt{2x - 1}\] Squaring both sides, we get : \[0 = 2x -1\] Solving for \(x\), we therefore find that \(x = 0.5\) So, the x-intercept here is \(0.5\).
02

Find the y-intercept

Attempt to find the y-intercept by setting \(x = 0\) in our function \[ y = \sqrt{2*0 - 1}\] Since we cannot take the square root of a negative number in the real number system, there is no valid y-intercept.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the x-intercept
The x-intercept is the point where a graph crosses the x-axis. At this point, the y-coordinate is zero. To find the x-intercept of an equation, substitute 0 for y and solve for x. For example, in the equation \(y=\sqrt{2x-1}\), you set \(y=0\) to find where the graph crosses the x-axis. When \(y=0\), the equation becomes \(0=\sqrt{2x-1}\).
By squaring both sides of the equation, you eliminate the square root, leaving you with \(0=2x-1\). Solving this linear equation for x, you add 1 to both sides, obtaining \(2x=1\), and then divide by 2, giving \(x=0.5\).
Therefore, the x-intercept of the function is 0.5. This tells us that the graph crosses the x-axis at the point (0.5, 0). This is an essential concept because identifying the x-intercept helps in sketching the graph of the function.
Exploring the y-intercept
The y-intercept is the point where a graph crosses the y-axis. At this point, the x-coordinate is zero. To find the y-intercept of an equation, set x to 0 and solve for y. In the case of the equation \(y=\sqrt{2x-1}\), by substituting \(x=0\), the equation becomes \(y=\sqrt{2\times0-1}\), which simplifies to \(y=\sqrt{-1}\).
Since \(\sqrt{-1}\) is not a real number, there is no real solution for y, indicating that the graph does not cross the y-axis. This situation is crucial to understand because not all functions will have a y-intercept within the realm of real numbers. Recognizing when a graph does not have a y-intercept can help in accurately sketching and interpreting the behavior of the graph.
Insight into the square root function
A square root function is characterized by the radical symbol \(\sqrt{}\) and is defined only for non-negative values of the expression inside the square root. This is because square roots of negative numbers are not real. For example, in \(y=\sqrt{2x-1}\), the domain is determined by the condition that the expression inside the square root, \(2x-1\), must be greater than or equal to zero.
Solving \(2x-1\geq0\) gives \(x\geq0.5\). This means the function is defined only for values of \(x\) that satisfy this inequality. Any \(x\) less than 0.5 would make \(2x-1\) negative, resulting in an imaginary number. This restriction affects the graph as well, starting at \(x=0.5\) and continuing to the right. Understanding the behavior and restrictions of square root functions is important when dealing with graphing and real-world applications, ensuring correct interpretation and utilization.

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