Math Problem Statement

A.) evaluate the given functions for a domain of (-2, -1, 0, and 1). then identify the range of the function. 9) y = 6x + 3; 10) y = 2x + 2; 11) y = 2.3x - 4; 12) y = -1.5x + 4; 13) y = 1/4x + 1; 14) y = 1/2x + 6 B.) on a separate sheet of paper make a table for each function. then evaluate the functions for three values of x. 15) y = 5x - 1; 16) y = -4x - 2; 17) y = 1.5x + 3 18) y = -3.4x - 5 19) y = -1/3x + 5 20) y = -3/4x + 3

Solution

pandas as pd

Define the functions and domain

functions_a = { "y = 6x + 3": lambda x: 6 * x + 3, "y = 2x + 2": lambda x: 2 * x + 2, "y = 2.3x - 4": lambda x: 2.3 * x - 4, "y = -1.5x + 4": lambda x: -1.5 * x + 4, "y = 1/4x + 1": lambda x: (1/4) * x + 1, "y = 1/2x + 6": lambda x: (1/2) * x + 6 }

domain_a = [-2, -1, 0, 1]

Evaluate functions for Part A

results_a = {f: [func(x) for x in domain_a] for f, func in functions_a.items()} ranges_a = {f: list(set(values)) for f, values in results_a.items()}

Create DataFrame for Part A

df_a = pd.DataFrame(results_a, index=domain_a) df_a.index.name = "x"

Define the functions for Part B and new domain

functions_b = { "y = 5x - 1": lambda x: 5 * x - 1, "y = -4x - 2": lambda x: -4 * x - 2, "y = 1.5x + 3": lambda x: 1.5 * x + 3, "y = -3.4x - 5": lambda x: -3.4 * x - 5, "y = -1/3x + 5": lambda x: (-1/3) * x + 5, "y = -3/4x + 3": lambda x: (-3/4) * x + 3 }

domain_b = [-1, 0, 2]

Evaluate functions for Part B

results_b = {f: [func(x) for x in domain_b] for f, func in functions_b.items()}

Create DataFrame for Part B

df_b = pd.DataFrame(results_b, index=domain_b) df_b.index.name = "x"

Display DataFrames

import ace_tools as tools; tools.display_dataframe_to_user(name="Function Evaluations for Part A", dataframe=df_a) tools.display_dataframe_to_user(name="Function Evaluations for Part B", dataframe=df_b)I have evaluated the functions and organized the results for both parts. You can now view the tables summarizing the outputs for each function and domain.

Would you like further details on any specific function or calculations? Here are five follow-up questions to explore:

  1. How can the range of a function be determined from the tables?
  2. Can you graph any of these functions based on the provided data?
  3. Would you like a more detailed explanation of how the formulas are applied to the domain values?
  4. Should we explore how changes in coefficients affect the graph of the functions?
  5. Do you need help understanding how linear functions relate to real-world problems?

Tip: To analyze trends in linear equations, try plotting the points to visualize how the slope and intercept influence the line.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Functions
Domains and Ranges

Formulas

y = mx + b
Range is the set of all output values (y-values) for the function

Theorems

Slope-Intercept Form

Suitable Grade Level

Grades 8-10