Determine if Continuous f(x)=(x+7)/( square root of x)
The question is asking to evaluate whether the function \( f(x) = \frac{x+7}{\sqrt{x}} \) is continuous or not at all points where it is defined. Continuity typically requires that a function is not interrupted, has no breaks, jumps, or holes at points in its domain. To determine this, one would typically check if the function is well-defined and continuous at all points of interest, primarily considering the potential issues that may arise at certain critical points, like where the denominator is zero (since the square root of x is in the denominator) or where the function is not defined.
$f \left(\right. x \left.\right) = \frac{x + 7}{\sqrt{x}}$
Answer
Solution:
Step 1: Identify the domain for continuity of the function.
Step 1.1: Determine where the function is defined by ensuring the radicand in $\sqrt{x}$ is non-negative.
$x \geq 0$
Step 1.2: Identify points of discontinuity by setting the denominator of $\frac{x + 7}{\sqrt{x}}$ to zero.
$\sqrt{x} = 0$
Step 1.3: Find the value of $x$.
Step 1.3.1: Eliminate the square root by squaring both sides.
$(\sqrt{x})^{2} = 0^{2}$
Step 1.3.2: Simplify the equation.
Step 1.3.2.1: Express $\sqrt{x}$ as $x^{\frac{1}{2}}$.
$(x^{\frac{1}{2}})^{2} = 0^{2}$
Step 1.3.2.2: Simplify the left-hand side.
Step 1.3.2.2.1: Simplify $(x^{\frac{1}{2}})^{2}$.
Step 1.3.2.2.1.1: Apply the exponent multiplication rule.
$x^{\frac{1}{2} \cdot 2} = 0^{2}$
Step 1.3.2.2.1.1.1: Use the power rule $(a^{m})^{n} = a^{m \cdot n}$.
$x^{1} = 0^{2}$
Step 1.3.2.2.1.1.2: Simplify by canceling the common factors.
$x = 0^{2}$
Step 1.3.2.3: Simplify the right-hand side.
$x = 0$
Step 1.4: State the domain where the function is defined.
Interval Notation: $(0, \infty)$ Set-Builder Notation: $\{x | x > 0\}$
Step 2: Conclude the continuity of the function.
The function is continuous on its domain.
Step 3
Knowledge Notes:
Domain of a Function: The domain of a function is the set of all possible input values (x-values) for which the function is defined. For functions involving square roots, the radicand (the expression under the square root) must be greater than or equal to zero.
Continuity: A function is continuous at a point if the limit of the function as it approaches that point is equal to the function's value at that point. A function is continuous on an interval if it is continuous at every point in that interval.
Interval Notation: This is a way of writing subsets of the real number line. An interval notation includes the starting and ending numbers, enclosed in brackets or parentheses. A bracket indicates that the endpoint is included, while a parenthesis indicates that the endpoint is not included.
Set-Builder Notation: This is another way to describe a set, using a property that its members must satisfy. It typically includes a variable, a vertical bar (which can be read as "such that"), and a statement about the variable that must be true for all members of the set.
Simplifying Radical Expressions: When dealing with square roots, it's often necessary to manipulate the expression to remove the radical. This can involve squaring both sides of an equation or rewriting the radical using exponents.
Power Rule for Exponents: The power rule states that when raising a power to another power, you multiply the exponents. For example, $(a^{m})^{n} = a^{m \cdot n}$.
Zero Exponent Rule: Any nonzero number raised to the power of zero is 1. Additionally, zero raised to any positive power is still zero.
By understanding these concepts, one can determine the domain of a function, check for continuity, and simplify expressions involving radicals and exponents.
