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 \( f(2)+f(3) \neq f(5) \). 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?

**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.