5 from Unit 5, Lesson 4 Function $$ f $$ is defined by the equation $$ f(x)=x^{2} $$. a. What is $$ f(2) $$ ? b. What is $$ f(3) $$ ? c. Explain why $$ f(2)+f(3)
eq f(5) $$. 6 from Unit 4, Lesson 2 Priya bought two plants for a science experiment. When she brought them home, the fil plant was 5 cm tall, and the second plant was 4 cm . Since then, the first plant has grown cm a week, and the second plant has grown 0.75 cm a week. a. Which plant is taller at the end of 2 weeks? Explain your reasoning. b. Which plant is taller at the end of 10 weeks? Explain your reasoning. c. Priya represents this situation with the equation $$ 5+0.5 w=4+0.75 w $$, where $$ w $$ represents the end of week $$ w $$. What does the solution to this equation, $$ w=4 $$ represent in this situation? d. What do the solutions to the inequality $$ 5+0.5 w4+0.75 w $$ represent in this situation?

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5 from Unit 5, Lesson 4 Function $$ f $$ is defined by the equation $$ f(x)=x^{2} $$. a. What is $$ f(2) $$ ? b. What is $$ f(3) $$ ? c. Explain why $$ f(2)+f(3) 
eq f(5) $$. 6 from Unit 4, Lesson 2 Priya bought two plants for a science experiment. When she brought them home, the fil plant was 5 cm tall, and the second plant was 4 cm . Since then, the first plant has grown cm a week, and the second plant has grown 0.75 cm a week. a. Which plant is taller at the end of 2 weeks? Explain your reasoning. b. Which plant is taller at the end of 10 weeks? Explain your reasoning. c. Priya represents this situation with the equation $$ 5+0.5 w=4+0.75 w $$, where $$ w $$ represents the end of week $$ w $$. What does the solution to this equation, $$ w=4 $$ represent in this situation? d. What do the solutions to the inequality $$ 5+0.5 w>4+0.75 w $$ represent in this situation?

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**Problem 5** Given the function $$ f(x)=x^2 $$: a. Compute $$ f(2) $$: $$ f(2)=2^2=4 $$ b. Compute $$ f(3) $$: $$ f(3)=3^2=9 $$ c. Explain why $$ f(2)+f(3) \neq f(5) $$: We have: $$ f(2)+f(3)=4+9=13 $$ and $$ f(5)=5^2=25 $$ Since $$ 13 \neq 25 $$, the equation $$ f(2)+f(3)=f(5) $$ does not hold. This shows that in general, for a function like $$ f(x)=x^2 $$, it is not true that the sum of the images equals the image of the sum. --- **Problem 6** Let the first plant's height be represented by: $$ h_1(w)=5+0.5w $$ and the second plant's height be represented by: $$ h_2(w)=4+0.75w, $$ where $$ w $$ is the number of weeks. a. **Which plant is taller at the end of 2 weeks?** Calculate the heights at $$ w=2 $$: $$ h_1(2)=5+0.5(2)=5+1=6 \quad \text{cm} $$ $$ h_2(2)=4+0.75(2)=4+1.5=5.5 \quad \text{cm} $$ Since $$ 6 > 5.5 $$, the first plant is taller at the end of 2 weeks. b. **Which plant is taller at the end of 10 weeks?** Calculate the heights at $$ w=10 $$: $$ h_1(10)=5+0.5(10)=5+5=10 \quad \text{cm} $$ $$ h_2(10)=4+0.75(10)=4+7.5=11.5 \quad \text{cm} $$ Since $$ 11.5 > 10 $$, the second plant is taller at the end of 10 weeks. c. **Interpret the equation $$ 5+0.5w=4+0.75w $$ and its solution $$ w=4 $$:** The equation $$ 5+0.5w=4+0.75w $$ represents the week when both plants have the same height. Solving for $$ w $$: $$ 5+0.5w=4+0.75w $$ Subtract $$ 4 $$ from both sides: $$ 1+0.5w=0.75w $$ Subtract $$ 0.5w $$ from both sides: $$ 1=0.25w $$ Divide both sides by $$ 0.25 $$: $$ w=4 $$ Thus, $$ w=4 $$ means that at the end of 4 weeks, both plants are of equal height. d. **Interpret the inequality $$ 5+0.5w>4+0.75w $$ and its solution:** Solve the inequality: $$ 5+0.5w>4+0.75w $$ Subtract $$ 4 $$ from both sides: $$ 1+0.5w>0.75w $$ Subtract $$ 0.5w $$ from both sides: $$ 1>0.25w $$ Divide both sides by $$ 0.25 $$: $$ w<4 $$ This inequality indicates that for $$ w<4 $$ (i.e., before 4 weeks pass), the height of the first plant is greater than the height of the second plant. **Problem 5** Given $$ f(x) = x^2 $$: a. $$ f(2) = 4 $$ b. $$ f(3) = 9 $$ c. $$ f(2) + f(3) = 13 $$ but $$ f(5) = 25 $$, so $$ f(2) + f(3) \neq f(5) $$. --- **Problem 6** Plants grow as: - First plant: $$ h_1(w) = 5 + 0.5w $$ - Second plant: $$ h_2(w) = 4 + 0.75w $$ a. After 2 weeks: - First plant: 6 cm - Second plant: 5.5 cm - First plant is taller. b. After 10 weeks: - First plant: 10 cm - Second plant: 11.5 cm - Second plant is taller. c. Equation $$ 5 + 0.5w = 4 + 0.75w $$ shows when both plants are equal. Solving gives $$ w = 4 $$ weeks. d. Inequality $$ 5 + 0.5w > 4 + 0.75w $$ shows that the first plant is taller before 4 weeks.

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