Problem 37 Let \(f(x)=x^{2} .\) Find \(x\) ... [FREE SOLUTION]

Chapter 11: Problem 37

Let \(f(x)=x^{2} .\) Find \(x\) such that \(f(x)=19\).

Short Answer

Expert verified

x = \sqrt{19} or x = -\sqrt{19}.

Step by step solution

- Understanding the Problem

We are given the function \(f(x) = x^2\) and need to find the value of \(x\) such that \(f(x) = 19\). This means we need to solve the equation \(x^2 = 19\).

- Setting Up the Equation

Start by equating the function to 19: \[x^2 = 19\].

- Solving the Equation

To find \(x\), take the square root of both sides of the equation: \[x = \pm \sqrt{19}\].

- Simplifying the Solution

Simplify the square root to get the final solution: \[x = \sqrt{19} \text{ or } x = -\sqrt{19}\].

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Key Concepts

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

function notation

In mathematics, functions are a way to show a relationship between two sets of numbers or objects. Function notation, such as \(f(x)\), is a compact way to represent functions. Here, \(f\) is the name of the function, and \(x\) is the input value. The function \(f(x) = x^2\) tells us that for any input \(x\), the output is \(x\) squared. This is crucial for solving problems because it allows us to understand how the input is transformed into the output. Using function notation, we can easily see that if we want \(f(x)\) to be 19, we need to find an input \(x\) that, when squared, equals 19.

square roots

Square roots help us reverse the process of squaring a number. The square root of a number \(n\), written as \(\sqrt{n}\), is a value that, when multiplied by itself, gives \(n\). For example, \(\sqrt{4} = 2\) because \(2 \times 2 = 4\). When solving the equation \(x^2 = 19\), we take the square root of both sides to find \(x\). This gives us two solutions: \(x = \sqrt{19}\) and \(x = -\sqrt{19}\), because both \((\sqrt{19})^2\) and \((-\sqrt{19})^2\) equal 19.

equations

Equations are statements that show the equality between two expressions. In this exercise, the equation \(x^2 = 19\) equates the square of \(x\) to 19. To solve this quadratic equation, we need to find values of \(x\) that satisfy the equation. By taking the square root of both sides, we solve for \(x\). The two solutions, \(x = \sqrt{19}\) and \(x = -\sqrt{19}\), show that both positive and negative values can be solutions. Understanding how to manipulate equations is vital for solving many types of problems in mathematics.

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91影视

Let \(f(x)=x^{2} .\) Find \(x\) such that \(f(x)=19\).

Short Answer

Step by step solution

- Understanding the Problem

- Setting Up the Equation

- Solving the Equation

- Simplifying the Solution

Key Concepts

function notation

square roots

equations

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