Question: A plant biologist models the growth efficiency of a drought-resistant crop with $ G(t) = rac{t^2 - 4}{t - 2} $. Simplify $ G(t) $ and determine its domain.
["Simplifying Plant Growth Efficiency: A Mathematical Breakdown of $ G(t) = \frac{t^2 - 4}{t - 2} $", "Understanding plant growth efficiency is crucial in developing drought-resistant crops, and mathematical modeling often plays a key role in analyzing biological productivity. A key function used in such models is $ G(t) = \dfrac{t^2 - 4}{t - 2} $, which represents the efficiency as a function of time $ t $. In this article, we simplify the expression and clarify its domain—insights valuable for plant biologists and agricultural researchers.", "---", "### What is $ G(t) $?", "The function $ G(t) = \dfrac{t^2 - 4}{t - 2} $ models growth efficiency based on a variable such as time or resource availability. While at first glance it appears straightforward, simplifying the expression reveals deeper biological interpretation and reveals constraints essential for real-world application.", "---", "### Step 1: Simplify the Expression", "Notice that the numerator $ t^2 - 4 $ is a difference of squares:", "$$\nt^2 - 4 = (t - 2)(t + 2)\n$$", "So, substitute back into $ G(t) $:", "$$\nG(t) = \dfrac{(t - 2)(t + 2)}{t - 2}\n$$", "For all $ t <br/>\neq 2 $, the $ t - 2 $ terms cancel:", "$$\nG(t) = t + 2\n$$", "Thus, the simplified form is:", "$$\nG(t) = t + 2, \quad \ ext{provided } t <br/>\ne 2\n$$", "---", "### Step 2: Determine the Domain", "The original expression $ G(t) = \dfrac{t^2 - 4}{t - 2} $ is undefined wherever the denominator is zero. Setting $ t - 2 = 0 $ gives:", "$$\nt = 2\n$$", "Although $ G(t) = t + 2 $ for all $ t <br/>\ne 2 $, the function is undefined at $ t = 2 $. Therefore, the domain excludes this point.", "Domain:\n$$\nt \in \mathbb{R} \setminus {2}\n$$\nor in interval notation:\n$$\n(-\infty, 2) \cup (2, \infty)\n$$", "---", "### Biological Significance", "For plant biologists modeling drought-resistant crops, the simplified linear model $ G(t) = t + 2 $ suggests growth efficiency increases linearly over time—useful for predicting performance under stress. However, the exclusion at $ t = 2 $ may indicate a critical transition point—such as a developmental stage or environmental threshold—where the simplified model no longer applies. Recognizing the domain helps avoid misinterpretation during long-term field trials or simulation studies.", "---", "### Conclusion", "Simplifying $ G(t) = \dfrac{t^2 - 4}{t - 2} $ yields $ G(t) = t + 2 $ for $ t <br/>\ne 2 $, reflecting efficient, predictable growth loss-free up to the critical time $ t = 2 $. Understanding both the algebraic simplification and domain constraints enables precise modeling of drought-resistant crops, supporting better decision-making in agricultural science.", "---", "Keywords:\nplant biologist, drought-resistant crops, growth efficiency, G(t) function, simplify rational expression, domain of G(t), $ t^2 - 4 $, $ t - 2 $, $ t + 2 $, algebra in biology, modeling plant growth"]
