Question

Does the graph of the function y = 3^3√x−1 show that the function is increasing or decreasing.
neither it is consistent
decreasing
increasing

Answer 1

To determine whether the function \( y = 3\sqrt[3]{x} - 1 \) is increasing or decreasing, we can analyze its derivative. The function can be rewritten as \( y = 3x^{1/3} - 1 \).

### Finding the Derivative

1. Differentiate the function:
\[
y' = \frac{d}{dx}(3x^{1/3} - 1)
\]
Using the power rule:
\[
y' = 3 \cdot \frac{1}{3} x^{-2/3} = x^{-2/3}
\]

### Analyzing the Derivative

The derivative \( y' = x^{-2/3} \) is defined for all \( x > 0 \) (it approaches infinity as \( x \) approaches zero from the right).

- For \( x > 0 \), \( y' \) is positive because \( x^{-2/3} > 0 \).
- For \( x < 0 \), the function \( \sqrt[3]{x} \) is defined, and \( y' \) will still be positive (since \( x^{-2/3} \) is positive for negative \( x \) values).

### Conclusion

Since the derivative \( y' \) is positive for all values in the domain of the function, we conclude that:

- The function \( y = 3\sqrt[3]{x} - 1 \) is **increasing** on its entire domain.

Thus, the graph of the function shows that it is increasing.