Problem 38 - Calculus of a Single Variable

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The roots of \(\sqrt{f(x)}=0\) coincide with the roots of \(f(x)=0\).

Step-by-Step Solution

Answer

The statement is true. The roots of the two given equations, \(\sqrt{f(x)}=0\) and \(f(x)=0\), do coincide.

Step 1: Understand the statement

The statement is suggesting that the roots of the sqrt(f(x))=0 equation are the same as the roots of the f(x)=0 equation. Roots are the values of x that make the equation true (equal to zero).

Step 2: Find roots of \(f(x) = 0\)

To find the roots of the f(x)=0 equation, we are essentially setting f(x) equal to zero and solving for x. For a general function, f(x)=0, the roots depend on the function itself. But regardless of the function, if f(x)=0, then x is a root of the function.

Step 3: Find roots of \(\sqrt{f(x)} = 0\)

To find the roots of the \(\sqrt{f(x)} = 0\) equation, we set \(\sqrt{f(x)}\) equal to zero, which implies that f(x) must be equal to zero since the square root of zero is zero. Therefore, the roots of \(\sqrt{f(x)} = 0\) are the same as those for f(x) = 0

Step 4: Comparing the roots

Since both types of equations lead to x values that make f(x) equal to zero, the statement in the problem is true. The roots of \(\sqrt{f(x)} = 0\) do indeed coincide with the roots of f(x) = 0.

Key Concepts

Square Root Function

The square root function involves taking the square root of a given expression. This mathematical operation, denoted by the square root symbol (\(\sqrt{}\)), essentially asks: "What number, multiplied by itself, gives this value?" Let's look at an example: the square root of 9 is 3, because 3 times 3 equals 9. When dealing with equations, the square root function allows us to determine certain values for variables that will satisfy the equation.
In equations like \(\sqrt{f(x)}=0\), we're looking for the points where \(f(x)\) becomes zero. This is because the only number whose square root is zero, is zero itself. Understanding how this function works is key to solving equations that involve square roots.

Zero of a Function

The zero of a function refers to those specific values of \(x\) that make the function equal zero. In simpler terms, it is where the graph of the function crosses the x-axis. To find the zero of a function, one must solve the equation by setting \(f(x) = 0\) and finding the values of \(x\) that satisfy this condition.
For example, in the function \(f(x) = x - 3\), setting it to zero, \(x - 3 = 0\), gives the zero of the function at \(x = 3\). In our case, both \(f(x)=0\) and \(\sqrt{f(x)}=0\) will lead us to the same zeros, as they require the same condition: \(f(x) = 0\). Always remember that zeros are where functions become zero.

True or False Statements

True or false statements require us to assess the accuracy of given propositions. In mathematics, these kinds of statements are often used to test comprehension of properties and relations between equations. For problems like our exercise, the task is to analyze the statement by applying known mathematical rules and logic.
In the original problem, we assessed if the roots of \(\sqrt{f(x)}=0\) coincide with those of \(f(x)=0\). By finding that both expressions lead to \(f(x) = 0\) as a necessary condition, we can conclude that the statement is true. Such exercises are vital in developing critical thinking and deeper understanding of mathematical concepts.

Solving Equations

Solving equations is a fundamental skill in mathematics. The process involves manipulating an equation to find the value of the variable that makes the equation true. Here are some basic tips to solve equations effectively:

• Isolate the variable: Make the variable the subject of the equation.
• Simplify both sides: Break down complex expressions into simpler ones.
• Check your solution: Substitute your answer back into the original equation to verify.

In our problem, solving \(f(x) = 0\) involved identifying values of \(x\) that make the function zero. Similarly, solving \(\sqrt{f(x)} = 0\) led us to the same values since both require \(f(x)\) to be zero for their conditions to be satisfied. Understanding equation-solving techniques is crucial in mathematics, as it applies to numerous areas and functions, including the ones involving square roots.