Math  /  Algebra

QuestionIdentify which function is continuous for all xx in the interval (,)(-\infty, \infty) from the given options.

Studdy Solution
Assumptions. We are looking for a function that is continuous over the interval (,)(-\infty, \infty)2. A function is continuous at a point if the limit of the function as x approaches that point exists and is equal to the value of the function at that point.
3. A function is continuous over an interval if it is continuous at every point in that interval.

2We start by examining the first function f(x)=x10x2+6f(x)=\frac{x-10}{x^{2}+6}. This function is a rational function and is continuous everywhere except for the values of x that make the denominator zero.3We find the values of x that make the denominator zero.
x2+6=0x^{2}+6=04olve the equation for x.
There are no real solutions for this equation, so the function f(x)=x10x2+6f(x)=\frac{x-10}{x^{2}+6} is continuous everywhere.
5Next, we examine the second function f(x)=ln(x10)f(x)=\ln (x-10). The natural logarithm function is continuous for all positive values of x.6We find the values of x that make the argument of the logarithm positive.
x10>0x-10>07olve the inequality for x.
x>10x>10So, the function f(x)=ln(x10)f(x)=\ln (x-10) is not continuous everywhere, because it is not defined for x10x \leq10.
8Next, we examine the third function f(x)=xsin(x)f(x)=\frac{x}{-\sin (x)}. This function is a rational function and is continuous everywhere except for the values of x that make the denominator zero.9We find the values of x that make the denominator zero.
sin(x)=0-\sin (x)=010olve the equation for x.
The solutions for this equation are x=(2n+)π2x = (2n+)\frac{\pi}{2}, where nn is an integer. So, the function f(x)=xsin(x)f(x)=\frac{x}{-\sin (x)} is not continuous everywhere.
11Next, we examine the fourth function f(x)=x10f(x)=\sqrt{x-10}. The square root function is continuous for all nonnegative values of its argument.
12We find the values of x that make the argument of the square root nonnegative.
x100x-10 \geq013olve the inequality for x.
x10x \geq10So, the function f(x)=x10f(x)=\sqrt{x-10} is not continuous everywhere, because it is not defined for x<10x <10.
14Next, we examine the fifth function f(x)=x10x+6f(x)=\frac{x-10}{x+6}. This function is a rational function and is continuous everywhere except for the values of x that make the denominator zero.15We find the values of x that make the denominator zero.
x+6=0x+6=016olve the equation for x.
x=6x=-6So, the function f(x)=x10x+6f(x)=\frac{x-10}{x+6} is not continuous everywhere, because it is not defined for x=6x = -6.
17We have examined all the given functions and found that only the first function f(x)=x10x2+6f(x)=\frac{x-10}{x^{2}+6} is continuous everywhere.So, the function that is continuous everywhere is f(x)=x10x2+6f(x)=\frac{x-10}{x^{2}+6}.

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