Question

compare the graphs of f (x) = √x + 1 and g (x) = ∛x + 1. which one of the following options is true?
a. both functions go through the points (-1, 0) and (0, 1).
b. both functions have the same domain, x > 0.
c. both functions have the same range , (-∞, ∞).
d. Both functions are increasing on their domain.

Answer 1

Let's analyze the functions \( f(x) = \sqrt{x} + 1 \) and \( g(x) = \sqrt[3]{x} + 1 \) and evaluate the options provided.

1. **Domain:**
- The domain of \( f(x) = \sqrt{x} + 1 \) is \( x \geq 0 \) (since the square root function is only defined for non-negative \( x \)).
- The domain of \( g(x) = \sqrt[3]{x} + 1 \) is \( (-\infty, \infty) \) (since the cube root function is defined for all real \( x \)).
- Hence, both functions do **not** have the same domain.

2. **Range:**
- The range of \( f(x) = \sqrt{x} + 1 \) is \( [1, \infty) \) (the minimum value of \( \sqrt{x} \) is 0, so \( \sqrt{x} + 1 \) starts from 1 and goes to \( \infty \)).
- The range of \( g(x) = \sqrt[3]{x} + 1 \) is \( (-\infty, \infty) \) (since \( \sqrt[3]{x} \) can take any real value, adding 1 shifts the entire range but does not restrict it).
- Therefore, both functions do **not** have the same range.

3. **Points of Intersection:**
- For the point \((-1, 0)\):
- For \( f(-1) \): Not defined (since \(-1 < 0\)).
- For \( g(-1) = \sqrt[3]{-1} + 1 = -1 + 1 = 0\).
- Hence, \((-1, 0)\) is not on \( f(x) \).

- For the point \( (0, 1) \):
- For \( f(0) = \sqrt{0} + 1 = 1\).
- For \( g(0) = \sqrt[3]{0} + 1 = 0 + 1 = 1\).
- Thus, both functions go through the point \( (0, 1) \), but not through the point \( (-1, 0)\).
- Therefore, option **a** is **false**.

4. **Increasing Nature of Functions:**
- To determine if both functions are increasing, we can consider their derivatives:
- \( f'(x) = \frac{1}{2\sqrt{x}} \) for \( x > 0\) (positive, therefore \( f(x) \) is increasing).
- \( g'(x) = \frac{1}{3\sqrt[3]{x^2}} \) for all \( x \neq 0\) (positive for all \( x\), thus \( g(x) \) is also increasing).
- Since both functions are increasing (even though \( f(x) \) is defined for \( x \geq 0\) and \( g(x) \) for all \( x\)), option **d** is **true** concerning their nature on their respective domains.

### Conclusion:
The correct option is **d**: Both functions are increasing on their domain.