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Chapter 1: Problem 23

Finding the Zeros of a Function Find the zeros of the function algebraically. $$f(x)=\sqrt{2 x}-1$$

Short Answer

Expert verified
The zero of the function \(f(x) = \sqrt{2x} - 1\) is \(x = \frac{1}{2}\).

Step by step solution

01

Set the Function Equal to Zero

We begin by setting the function equal to zero. This gives us the equation: \(\sqrt{2x} - 1 = 0\)
02

Solve the Equation for x

Add 1 to both sides to isolate the square root term, then square both sides to eliminate the square root: \((\sqrt{2x} - 1 + 1)^2 = (0 + 1)^2 \Rightarrow (2x) = 1 \Rightarrow x = 1/2\
03

Check for Valid Solutions

Lastly, we should check if this solution is valid by substituting the value back into the given function: \(f(\frac{1}{2}) = \sqrt{2*(\frac{1}{2})} - 1 = 0 \), so \(x=\frac{1}{2}\) is a valid solution.

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Key Concepts

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

Solving Square Root Equations
Square root equations often intimidate students, but with the right approach, they can be solved quite smoothly. To tackle them, it's essential to isolate the square root on one side of the equation. Once isolation is achieved, the next step involves squaring both sides, which removes the square root and simplifies the equation to a more familiar algebraic form. It's important to proceed with caution, as squaring can introduce extraneous solutions. That's why you should always check your answers in the original equation.
Setting Functions Equal to Zero
Finding the zeros of a function is equivalent to solving the equation when the function is set equal to zero. This is because the zeros are the values of x where the function crosses or touches the x-axis, meaning the output of the function is zero. To effectively find the zeros, one should first rewrite the function in a way that the entire expression is equal to zero before proceeding with solving the equation. This approach gives you a clear target for what you're trying to find鈥攖he values of x that make the function equal to zero.
Algebraic Manipulation
Algebraic manipulation is the process used to transform an equation or expression into a different form, which often makes solving the problem simpler. It involves performing operations like addition, subtraction, multiplication, division, and factoring on both sides of an equation, with the goal of isolating the variable of interest. These techniques require a strong understanding of algebraic principles and the ability to apply them correctly. Incorrect manipulation can lead to wrong answers, so it's paramount to perform each step with precision and verify your work as you go.
Validating Solutions
After solving equations, especially those involving squared terms or higher degrees, there may be potential for extraneous solutions鈥攕olutions that emerge from the algebraic process but do not satisfy the original equation. To confirm the validity of a solution, you need to substitute it back into the original equation. This process is a crucial step to ensure that the provided answers truly represent the zeros of the function in question. Validating your solutions helps avoid the common mistake of reporting incorrect answers and is a mark of thorough mathematical practice.

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Most popular questions from this chapter

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