Which function goes to positive co most quickly as \( x \) Increases? A \( y=\log (x)+100 \) B \( y=8^{x-9}-3 \) C \( y=x^{2}+5 x+6 \) D \( y=3 x^{3}+4 x^{3}-11 x-6 \)

To determine which function goes to positive infinity the quickest as \( x \) increases, we need to analyze the growth rates of each function as \( x \) approaches infinity. 1. **Function A: \( y = \log(x) + 100 \)** - The logarithmic function grows very slowly compared to polynomial and exponential functions. As \( x \to \infty \), \( y \to \infty \), but at a slow rate. 2. **Function B: \( y = 8^{x-9} - 3 \)** - This is an exponential function. Exponential functions grow very quickly as \( x \) increases. As \( x \to \infty \), \( y \to \infty \) rapidly. 3. **Function C: \( y = x^2 + 5x + 6 \)** - This is a quadratic function. Quadratic functions grow faster than logarithmic functions but slower than exponential functions. As \( x \to \infty \), \( y \to \infty \). 4. **Function D: \( y = 3x^3 + 4x^3 - 11x - 6 \)** - This simplifies to \( y = 7x^3 - 11x - 6 \), which is a cubic function. Cubic functions grow faster than quadratic functions but slower than exponential functions. As \( x \to \infty \), \( y \to \infty \). Now, let's summarize the growth rates: - **Logarithmic function (A)**: Slowest growth - **Quadratic function (C)**: Faster than logarithmic - **Cubic function (D)**: Faster than quadratic - **Exponential function (B)**: Fastest growth Thus, the function that goes to positive infinity the quickest as \( x \) increases is: **B \( y = 8^{x-9} - 3 \)**.