Determine if Continuous f(x)=exp(- square root of x-1)
The question asks whether the function f(x) = exp(-√(x-1)) is continuous or not. To determine this, one would need to analyze the function and its behavior across its domain. Continuity of a function at a point means that the function is defined at that point, its limit exists at that point, and the limit at that point is equal to the function's value. The square root implies restrictions on the domain to avoid complex numbers, and the exponential function is inherently continuous, but the composition of both needs to be examined. The question requires an inspection of the possible continuity or discontinuity at x = 1, as well as a general look at the continuity for all x in its domain.
$f \left(\right. x \left.\right) = e x p \left(\right. - \sqrt{x - 1} \left.\right)$
Answer
Solution:
Determining Continuity of the Function $f(x) = \exp(-\sqrt{x - 1})$
Step 1: Identify the Domain of the Function
To ensure the function is continuous, we need to find the range of $x$ values for which the function is defined.
Step 1.1: Solve for the Domain
The square root function requires non-negative arguments, so set the inside of the square root to be non-negative:
$$x - 1 \geq 0$$
Step 1.2: Isolate $x$
Add $1$ to both sides to solve for $x$:
$$x \geq 1$$
Step 1.3: Express the Domain
The domain of $f(x)$ is the set of $x$ values that satisfy the inequality. In interval notation, this is:
$$[1, \infty)$$ In set-builder notation, it is:
$$\{ x | x \geq 1 \}$$
Step 2: Analyze Continuity Within the Domain
Given the domain, the function $f(x)$ is continuous for all $x$ in its domain.
Step 3: Conclusion
The function $f(x) = \exp(-\sqrt{x - 1})$ is continuous on its domain $[1, \infty)$.
Knowledge Notes:
To determine the continuity of a function, one must first establish its domain, which is the set of all input values for which the function is defined. For the function $f(x) = \exp(-\sqrt{x - 1})$, the domain is restricted by the square root, as the square root function is only defined for non-negative numbers. Therefore, the expression under the square root, known as the radicand, must be greater than or equal to zero.
The exponential function, $\exp(x)$, is continuous for all real numbers. Since the composition of continuous functions is also continuous, and the square root function is continuous wherever it is defined (for non-negative numbers), the function $f(x)$ will be continuous on its domain.
Interval notation is a way of writing subsets of the real number line. An interval such as $[1, \infty)$ includes all the numbers starting from $1$ and going to infinity. The square bracket, $[$, indicates that $1$ is included in the set, while the parenthesis, $)$, indicates that infinity is not a number that can be reached or included.
Set-builder notation is another way to describe a set, defining the properties that its members must satisfy. For example, $\{ x | x \geq 1 \}$ describes the set of all $x$ such that $x$ is greater than or equal to $1$.
In summary, the relevant knowledge points include understanding the domain of a function, the continuity of the exponential and square root functions, and how to express sets using interval and set-builder notations.
