Question

graph the linear function ( f(x)=\frac{1}{2}x - 7 ). identify if it is a constant function. give the domain and range. use the graphing tool to graph the equation

Explanation:

Step1: Analyze the function type

A constant function has the form \( f(x) = c \) where \( c \) is a constant (the function value does not depend on \( x \)). The given function is \( f(x)=\frac{1}{2}x - 7 \), which is in the form of a linear function \( y=mx + b \) (where \( m=\frac{1}{2} \) and \( b = - 7 \)), not a constant function since the coefficient of \( x \) is non - zero.

Step2: Find the domain of the linear function

For a linear function of the form \( f(x)=mx + b \), there are no restrictions on the values of \( x \) (we can plug in any real number for \( x \) and get a real number output). So the domain is all real numbers, which in interval notation is \( (-\infty,\infty) \).

Step3: Find the range of the linear function

Since the slope \( m=\frac{1}{2}
eq0 \), as \( x \) takes on all real values (from \( -\infty \) to \( \infty \)), \( y = f(x)=\frac{1}{2}x-7 \) will also take on all real values. For example, if \( x \) approaches \( \infty \), \( \frac{1}{2}x-7 \) approaches \( \infty \), and if \( x \) approaches \( -\infty \), \( \frac{1}{2}x - 7 \) approaches \( -\infty \). So the range is also all real numbers, or in interval notation \( (-\infty,\infty) \).

Answer:

• The function \( f(x)=\frac{1}{2}x - 7 \) is not a constant function.
• Domain: \( (-\infty,\infty) \)
• Range: \( (-\infty,\infty) \)