End Behavior Bc5501

1. **Problem Statement:** We are given functions $f$, $g$, and $h$ with some values and a graph. The function $g$ is defined as $$g(x) = 2^x - 5.$$ We need to determine which statement about the end behavior of these functions as $x \to \infty$ is true. 2. **Recall End Behavior Rules:** - For exponential functions like $g(x) = 2^x - 5$, as $x \to \infty$, $2^x \to \infty$, so $g(x) \to \infty$. - For functions approaching zero, the function values get closer and closer to zero as $x$ increases. - For functions approaching positive infinity, the function values grow without bound. 3. **Analyze $g(x)$:** Since $g(x) = 2^x - 5$, as $x \to \infty$, $2^x \to \infty$, so $$g(x) = 2^x - 5 \to \infty.$$ Therefore, $g$ does not approach zero; it approaches positive infinity. 4. **Analyze $f$ and $h$ (based on given info):** - The problem states $f$ and $h$ values are given in a table and graph, but no explicit formulas. - The statements suggest $f$ and $h$ might approach positive infinity or zero. 5. **Evaluate each statement:** - "As $x$ increases, $f$ and $h$ both approach positive infinity." Possible if both grow without bound. - "As $x$ increases, $g$ is the only function to approach 0." False, since $g(x) \to \infty$. - "As $x$ increases, $f$ and $g$ both approach positive infinity." Possible if $f$ grows without bound. - "As $x$ increases, $f$ is the only function to approach 0." Possible if $f$ tends to zero and others do not. 6. **Conclusion:** Since $g(x) = 2^x - 5$ grows exponentially and tends to positive infinity, the only true statement is: **"As the value of $x$ increases, $f$ and $g$ both approach positive infinity."**