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:
Statement 1:
The range of both functions is all real numbers.
- Both and 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 , .
- For , as , , and thus .
True.
Statement 3:
As , .
- For , as , , and thus , not .
False.
Statement 4:
The slope of is greater than the slope of .
- The slope of is , and the slope of is . Since , the slope of is not greater than the slope of .
False.
Statement 5:
The -intercept of is less than the -intercept of .
- The -intercept of is obtained by setting :
The -intercept of is obtained by setting :
Since , the -intercept of is greater than that of , not less.
False.
Correct Statements:
- 1. The range of both functions is all real numbers.
- 2. As , .
Would you like a deeper explanation of any point?
Here are 5 related questions to expand understanding:
- How do you calculate the range of a linear function?
- Why does the slope affect how quickly a function approaches infinity?
- What are the steps to find the -intercept of a linear function?
- How can you compare the growth rates of two linear functions?
- What is the significance of the slope sign in determining a function's behavior?
Tip: For linear functions, always check the slope () and -intercept () 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