Determine if Continuous f(x) = square root of x-7

Solution:

Step 1: Identify the Domain for Continuity

Step 1.1: Establish the Range of $x$ for Which the Function is Defined

Determine where the function $\sqrt{x - 7}$ is non-negative by solving $x - 7 \geq 0$.

Step 1.2: Solve the Inequality

Isolate $x$ by adding $7$ to both sides, resulting in $x \geq 7$.

Step 1.3: Express the Domain

The function is defined for $x$ values in the interval $\left[ 7 , \infty \right)$.

Using Interval Notation: $\left[ 7 , \infty \right)$ Using Set-Builder Notation: $\{ x | x \geq 7 \}$

Step 2: Conclude the Continuity of the Function

The function is continuous over its domain.

Step 3:

(No further steps required)

Knowledge Notes:

The problem at hand involves determining whether the function $f(x) = \sqrt{x - 7}$ is continuous. To do this, we must first understand what continuity means for a function. A function is continuous at a point if the limit of the function as it approaches the point is equal to the function's value at that point. A function is continuous over an interval if it is continuous at every point within that interval.

The first step in assessing the continuity of $f(x)$ is to find its domain, which is the set of all $x$ values for which the function is defined. Since $f(x)$ involves a square root, the radicand (the expression inside the square root) must be greater than or equal to zero to ensure that the function returns real numbers. We set up the inequality $x - 7 \geq 0$ and solve for $x$ to find the domain.

The solution to the inequality is $x \geq 7$, which means that the function is defined for all $x$ values greater than or equal to 7. We can express this domain using interval notation as $\left[ 7 , \infty \right)$, which includes 7 and all numbers greater than 7, or using set-builder notation as $\{ x | x \geq 7 \}$, which reads as "the set of all $x$ such that $x$ is greater than or equal to 7."

Given that the domain is established, we can conclude that the function $f(x)$ is continuous over its domain because the square root function is inherently continuous wherever it is defined. There are no breaks, jumps, or points of discontinuity within this domain.

Understanding the domain of functions, especially those involving square roots, is crucial in determining continuity. The square root function is continuous for all non-negative inputs, and thus, the continuity of $f(x)$ depends solely on its domain.