1. \( g(x)=-x^{3}+21 \)

To analyze the function \( g(x) = -x^3 + 21 \), we can explore various features such as its domain, range, intercepts, and critical points. Let's go through these step by step. ### Step 1: Domain The domain of a polynomial function is all real numbers. Therefore, the domain of \( g(x) \) is: \[ \text{Domain: } (-\infty, \infty) \] ### Step 2: Range Since \( g(x) \) is a cubic function with a negative leading coefficient, it will decrease without bound as \( x \) approaches positive infinity and increase without bound as \( x \) approaches negative infinity. The maximum value occurs at the vertex. To find the maximum value, we can find the critical points by taking the derivative and setting it to zero. ### Step 3: Finding Critical Points Let's calculate the derivative \( g'(x) \): \[ g'(x) = -3x^2 \] Setting the derivative equal to zero to find critical points: \[ -3x^2 = 0 \] This gives: \[ x = 0 \] ### Step 4: Evaluating \( g(x) \) at Critical Points Now, we evaluate \( g(x) \) at \( x = 0 \): \[ g(0) = -0^3 + 21 = 21 \] ### Step 5: Range Conclusion Since the function decreases to negative infinity as \( x \) moves away from 0 in both directions, the range of \( g(x) \) is: \[ \text{Range: } (-\infty, 21] \] ### Step 6: Intercepts - **Y-intercept**: This occurs when \( x = 0 \): \[ g(0) = 21 \quad \text{(Y-intercept: (0, 21))} \] - **X-intercepts**: Set \( g(x) = 0 \): \[ -x^3 + 21 = 0 \implies x^3 = 21 \implies x = \sqrt[3]{21} \] Thus, the x-intercept is: \[ \text{X-intercept: } \left(\sqrt[3]{21}, 0\right) \] ### Summary of Features - **Domain**: \( (-\infty, \infty) \) - **Range**: \( (-\infty, 21] \) - **Y-intercept**: \( (0, 21) \) - **X-intercept**: \( \left(\sqrt[3]{21}, 0\right) \) - **Critical Point**: Maximum at \( (0, 21) \) If you need further analysis or specific calculations, please let me know!