Math Problem Statement

Consider the linear functions:

$m(x)=5-x$ and $p(x)=\frac{12}{3}x-1$

Which of the following statements are true? Select all that apply.

Responses

The range of both functions is all real numbers. The range of both functions is all real numbers.

As $x$ approaches $\infty$ , $p\left(x\right)$ approaches $\infty$ . As x approaches infinity, p times x approaches infinity.

As $x$ approaches $\infty$ , $m\left(x\right)$ approaches $\infty$ . As x approaches infinity, m times x approaches infinity.

The slope of $m\left(x\right)$ is greater than the slope of $p\left(x\right)$ . The slope of m times x is greater than the slope of p times x.

The $x-$ intercept of $m\left(x\right)$ is less than the $x-$ intercept of $p\left(x\right)$ .

Solution

Let's analyze the given functions step by step.

The Functions:

  1. m(x)=5xm(x) = 5 - x
  2. p(x)=123x1=4x1p(x) = \frac{12}{3}x - 1 = 4x - 1

Statement 1:

The range of both functions is all real numbers.

  • Both m(x)m(x) and p(x)p(x) are linear functions. The output of a linear function is unbounded unless the domain is restricted. Therefore, the range of both functions is all real numbers.
    True.

Statement 2:

As xx \to \infty, p(x)p(x) \to \infty.

  • For p(x)=4x1p(x) = 4x - 1, as xx \to \infty, 4x4x \to \infty, and thus p(x)p(x) \to \infty.
    True.

Statement 3:

As xx \to \infty, m(x)m(x) \to \infty.

  • For m(x)=5xm(x) = 5 - x, as xx \to \infty, x-x \to -\infty, and thus m(x)m(x) \to -\infty, not \infty.
    False.

Statement 4:

The slope of m(x)m(x) is greater than the slope of p(x)p(x).

  • The slope of m(x)=5xm(x) = 5 - x is 1-1, and the slope of p(x)=4x1p(x) = 4x - 1 is 44. Since 1<4-1 < 4, the slope of m(x)m(x) is not greater than the slope of p(x)p(x).
    False.

Statement 5:

The xx-intercept of m(x)m(x) is less than the xx-intercept of p(x)p(x).

  • The xx-intercept of m(x)m(x) is obtained by setting m(x)=0m(x) = 0: 0=5x    x=5.0 = 5 - x \implies x = 5. The xx-intercept of p(x)p(x) is obtained by setting p(x)=0p(x) = 0: 0=4x1    x=14.0 = 4x - 1 \implies x = \frac{1}{4}. Since 5>145 > \frac{1}{4}, the xx-intercept of m(x)m(x) is greater than that of p(x)p(x), not less.
    False.

Correct Statements:

  • 1. The range of both functions is all real numbers.
  • 2. As xx \to \infty, p(x)p(x) \to \infty.

Would you like a deeper explanation of any point?
Here are 5 related questions to expand understanding:

  1. How do you calculate the range of a linear function?
  2. Why does the slope affect how quickly a function approaches infinity?
  3. What are the steps to find the xx-intercept of a linear function?
  4. How can you compare the growth rates of two linear functions?
  5. What is the significance of the slope sign in determining a function's behavior?

Tip: For linear functions, always check the slope (mm) and yy-intercept (bb) to quickly understand their behavior.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Slopes
Intercepts
Behavior of Functions at Infinity

Formulas

General form of a linear function: y = mx + b

Theorems

Range of a linear function is all real numbers for unrestricted domains.

Suitable Grade Level

Grades 9-10