Determine if Continuous f(x) = square root of 7x+21
The given question is asking to evaluate the continuity of a function defined by f(x) = √(7x+21). To determine if this function is continuous, one would typically examine whether the function is well-defined, unbroken, and has no discontinuities (such as holes, jumps, or vertical asymptotes) at all points in its domain. Specifically, since this involves a square root function, special attention should be given to the domain of the function, ensuring that the argument of the square root (7x+21) is non-negative.
$f \left(\right. x \left.\right) = \sqrt{7 x + 21}$
Answer
Solution:
Step:1
Identify the domain to check the continuity of the function.
Step:1.1
Ensure the inside of the square root, $\sqrt{7x + 21}$, is non-negative by setting the expression inside the square root to be greater than or equal to zero: $7x + 21 \geq 0$.
Step:1.2
Determine the values of $x$ that satisfy the inequality.
Step:1.2.1
Isolate the variable by subtracting $21$ from both sides: $7x \geq -21$.
Step:1.2.2
Proceed to divide the inequality by $7$ to solve for $x$.
Step:1.2.2.1
Apply division to both sides of the inequality: $\frac{7x}{7} \geq \frac{-21}{7}$.
Step:1.2.2.2
Simplify the inequality by reducing terms.
Step:1.2.2.2.1
Eliminate the common factor of $7$.
Step:1.2.2.2.1.1
Remove the $7$ from the numerator and denominator: $\frac{\cancel{7}x}{\cancel{7}} \geq \frac{-21}{7}$.
Step:1.2.2.2.1.2
Express $x$ as being divided by $1$: $x \geq \frac{-21}{7}$.
Step:1.2.2.3
Clarify the right side of the inequality.
Step:1.2.2.3.1
Compute the division of $-21$ by $7$: $x \geq -3$.
Step:1.3
Conclude the domain as all $x$ values that satisfy the function: Interval Notation: $[-3, \infty)$, Set-Builder Notation: $\{x | x \geq -3\}$.
Step:2
Affirm that the function is continuous over its domain.
Step:3
There is no further action required as the function is continuous over its domain.
Knowledge Notes:
To determine if a function $f(x)$ is continuous, we need to ensure that it is defined for all values in its domain and that there are no breaks, jumps, or holes in the graph of the function. For the function $f(x) = \sqrt{7x + 21}$, we must first find the domain by setting the radicand (the expression inside the square root) to be greater than or equal to zero, since the square root of a negative number is not defined in the set of real numbers.
The steps involve solving the inequality $7x + 21 \geq 0$ to find the values of $x$ for which the function is defined. This is done by isolating $x$ on one side of the inequality and simplifying the expression. Once the domain is established, we can say that the function is continuous if there are no other restrictions or discontinuities within that domain.
In interval notation, the domain is expressed as a range of values in square brackets (for inclusive values) or parentheses (for exclusive values). In set-builder notation, the domain is expressed as a set of values that satisfy a given property, typically written as $\{x | \text{property of } x\}$.
For square root functions, the function is continuous over its domain as long as the radicand is non-negative. Since the domain of $f(x) = \sqrt{7x + 21}$ is $x \geq -3$, and there are no other restrictions, the function is continuous over the interval $[-3, \infty)$.
