Determine the domain and range of the function f(x) = 5x^4 - 10.
Check the final answer first, then review the worked steps.
Problem
Determine the domain and range of the function f(x) = 5x^4 - 10.
Answer
Domain: $(-\infty, \infty)$, Range: $[-10, \infty)$
Step-by-step solution
- Determine the Domain: The function $f(x) = 5x^4 - 10$ is a polynomial function. Polynomial functions are defined for all real numbers. Therefore, the domain of the function is all real numbers, which can be written in interval notation as $(-\infty, \infty)$.
- Determine the Range: To find the range, we need to consider the behavior of the function. The term $x^4$ is always non-negative, i.e., $x^4 \ge 0$ for all real numbers $x$. Multiplying by 5, we get $5x^4 \ge 0$. Subtracting 10 from both sides, we have $5x^4 - 10 \ge -10$. This means that the minimum value of the function is -10, and it can take any value greater than or equal to -10. Therefore, the range of the function is $[-10, \infty)$.