Determine if Continuous f(x)=( square root of x^2-25)/( natural log of x+9)
The problem involves examining the mathematical function f(x) = √(x^2 - 25) / (ln(x + 9)) to determine if it is continuous. Continuity of a function at a point means that the function is defined at that point, the limit of the function as it approaches the point exists, and the limit equals the function's value at that point. The question likely asks to investigate the behavior of the function across its entire domain, identify any points of discontinuity if they exist, and validate its continuity at points where it appears to be continuous by satisfying the definition of continuity.
Identify the domain to assess continuity of the function.
Ensure the argument inside
Isolate
The expression under the square root,
Find the values of
Balance the inequality by adding
Remove the square by taking the square root of both sides.
Simplify the resulting equation.
Extract
Simplify the square root of
Express
Extract
Recognize that absolute value represents the distance from zero.
Express
For the positive domain, consider
Without the absolute value, for positive
For the negative domain, consider
For negative
Combine as a piecewise condition.
The intersection of
Invert the inequality
Inverting the inequality gives
Simplify to find
Combine the solutions to get the domain.
Set the denominator of
Solve for
Use the properties of logarithms to rewrite the equation.
Convert the logarithmic equation to exponential form.
Isolate
Rewrite as
Since any number raised to zero is one,
Rearrange to solve for
Subtract
Calculate to find
The domain is the set of
Interval Notation:
The function
The problem involves determining the continuity of a function defined by
The argument of the natural logarithm,
The expression under the square root,
After finding the domain, we check for any points where the function might not be continuous. This includes points where the denominator is zero since division by zero is undefined.
The solution involves several mathematical concepts:
Inequalities: Solving inequalities to find the domain of the function.
Absolute value: Understanding that
Logarithms and exponentials: Using properties of logarithms and exponentials to solve equations involving the natural logarithm.
Continuity: Recognizing that a function is continuous if it is defined and does not have any breaks, jumps, or holes in its graph.
The final step is to express the domain in both interval notation and set-builder notation, which are two ways to describe sets of numbers. Interval notation uses intervals to describe the domain, while set-builder notation uses a condition that elements must satisfy to be included in the set.