Question

Find the domain of \(f(x)=\sqrt{10-2 x}\).

Short answer

The domain of the function is \[ (-\infty, 5] \].

Step-by-step solution

Understand the function

The function given is a square root function, specifically: \[ f(x) = \sqrt{10 - 2x} \]. We need to find the domain, which includes all possible values of \(x\) that make the expression inside the square root non-negative.

Set the inequality

For the square root to be defined, the expression inside must be greater than or equal to zero. Set up the inequality: \[ 10 - 2x \geq 0 \].

Solve for x

Solve the inequality for \(x\): \[ 10 - 2x \geq 0 \] Subtract 10 from both sides: \[ -2x \geq -10 \] Divide both sides by -2 (remember to reverse the inequality sign when dividing by a negative number): \[ x \leq 5 \].

Write the domain

The solution to the inequality \( x \leq 5 \) represents the domain of the function. Therefore, the domain is: \[ (-\infty, 5] \].

Key concepts

Inequality Solving

When faced with finding the domain of a function involving a square root, we need to ensure that the value inside the square root is non-negative. This leads us to set up an inequality. In this case, for the function \( f(x) = \sqrt{10 - 2x} \), we need the inside of the square root (\(10 - 2x\)) to be greater than or equal to zero.

Here are the steps to solve such an inequality:

• Write down the inequality: \(10 - 2x \geq 0\)
• Isolate the variable: Subtract 10 from both sides to get \(-2x \geq -10\)
• Divide by \(-2\): Remember, when you divide by a negative number, you must reverse the inequality sign, giving us \(x \leq 5\)

So the set of all values making \(10 - 2x\) non-negative is \(x \leq 5\). Hence, this inequality helps us determine the valid range for \(x\).

The key takeaway is to remember the rule about reversing the inequality sign when dividing or multiplying by a negative number. This is crucial for finding correct solutions.

Square Root Functions

Square root functions, like \( f(x) = \sqrt{10 - 2x} \), always involve ensuring the expression inside the root is non-negative, because the square root of a negative number is not defined in the set of real numbers.

In practical terms, it means:

• The expression inside the square root must be greater than or equal to zero.
• This creates an inequality that you need to solve.

For instance, we need \(10 - 2x\) to be non-negative for \( f(x)\) to be defined. By solving the inequality \(10 - 2x \geq 0\), we ensure that the square root function produces valid real number outputs.

Understanding and working with square root functions is essential because they appear frequently in various mathematical contexts and real-world applications. Grasp the concept well, and you'll find these problems easier to solve.

Function Domain

The domain of a function indicates all possible input values (\(x\)) for which the function produces a valid output. For the square root function \( f(x) = \sqrt{10 - 2x} \), our task is identifying all \(x\) values that result in a non-negative number inside the square root.

Here’s how we identify the domain:

• Write the condition for the expression inside the square root to be non-negative: \(10 - 2x \geq 0\)
• Solve the inequality to find the allowed \(x\) values: \(x \leq 5\)

So the domain of the function \( f(x) = \sqrt{10 - 2x} \) is \( (-\infty, 5] \). This means any real number less than or equal to 5 is included in the domain.

Understanding the domain is vital since it tells us the set of all acceptable inputs into our function, ensuring we only work with values that produce valid, real-valued outputs. Knowing this, you can confidently approach and solve these types of problems.