To determine the range of the function f(x) = x^5 + x – 1, we start by analyzing its behavior as x approaches positive and negative infinity.
As x approaches infinity, the term x^5, which is the leading term, dominates. Therefore, f(x) approaches infinity:
f(x) → ∞ as x → ∞
Conversely, as x approaches negative infinity, x^5 tends towards negative infinity:
f(x) → -∞ as x → -∞
This suggests that the function could take all real values. To verify this, we check if the function is continuous and if it is increasing or decreasing by finding its derivative:
f'(x) = 5x^4 + 1
This derivative is always positive since 5x^4 is non-negative and adding 1 ensures it never reaches zero. Thus, f(x) is strictly increasing over all real numbers.
Since the function is continuous and strictly increasing, it indicates that f(x) indeed takes all values from negative infinity to positive infinity.
In conclusion, the range of the function f(x) = x^5 + x – 1 is all real numbers, expressed in interval notation as:
Range: (-∞, ∞)