Let's analyze the given functions to determine which one is not continuous on the interval $(- \infty, \infty)$:

Option A: $f(x) = 4x^2 - 2x + 1$

This is a quadratic function (a polynomial). Polynomial functions are continuous everywhere on $(- \infty, \infty)$. So, $f(x)$ is continuous.

Option B: $g(x) = \frac{1}{x^3 + 3x^2 - 2x - 5}$

This is a rational function, which is continuous everywhere except where its denominator equals zero. To find where it is discontinuous, we need to solve for the values of $x$ that make the denominator zero: $x^3 + 3x^2 - 2x - 5 = 0$ The roots of this cubic equation would cause discontinuities. Since there are real values of $x$ that make the denominator zero, $g(x)$ is not continuous at those points. Therefore, $g(x)$ is not continuous on the interval $(- \infty, \infty)$.

Option C: $h(x) = \cos(\pi x)$

This is a cosine function, which is continuous everywhere since trigonometric functions like cosine are continuous on $(- \infty, \infty)$. Thus, $h(x)$ is continuous.

Option D: $k(x) = \frac{1}{e^x}$

This is an exponential function, and the exponential function $e^x$ is never zero for any real value of $x$, so the function is continuous on $(- \infty, \infty)$. Thus, $k(x)$ is continuous.

Conclusion:

The function $g(x) = \frac{1}{x^3 + 3x^2 - 2x - 5}$ is the one that is not continuous on the interval $(- \infty, \infty)$, due to points where the denominator equals zero.

Would you like more details on how to solve the cubic equation for the points of discontinuity?

Related Questions:

1. How do you find the points of discontinuity for a rational function?
2. What types of functions are guaranteed to be continuous on $(- \infty, \infty)$?
3. What is the graphical behavior near points of discontinuity?
4. How does the continuity of trigonometric functions differ from that of rational functions?
5. What methods are used to solve cubic equations?

Tip:

When analyzing the continuity of a rational function, always check where the denominator equals zero—those are the points of potential discontinuity.