Question

given $f(x)=\frac{1}{2}x + 4$, find the range.
$\circ$ $y\geq4$
$\circ$ $y\leq4$
$\circ$ $x\geq - 8$
$\circ$ all real numbers

Explanation:

Step1: Identify the function type

The function \( f(x)=\frac{1}{2}x + 4 \) is a linear function. The general form of a linear function is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. In this case, \( m=\frac{1}{2}\) and \( b = 4 \).

Step2: Analyze the range of a linear function

For a linear function with a non - zero slope (here \( m=\frac{1}{2}
eq0\)), as \( x \) takes on all real number values (the domain of a linear function \( y=mx + b\) is all real numbers), when \( x\) increases or decreases without bound, \( y=\frac{1}{2}x + 4\) will also take on all real number values.

• If \( x\to+\infty\), then \( y=\frac{1}{2}x + 4\to+\infty\) (since \(\frac{1}{2}x\) will become very large positive as \( x\) is large positive and adding 4 still keeps it large positive).
• If \( x\to-\infty\), then \( y=\frac{1}{2}x + 4\to-\infty\) (since \(\frac{1}{2}x\) will become very large negative as \( x\) is large negative and adding 4 does not change the fact that it goes to negative infinity).

So the range of the function \( f(x)=\frac{1}{2}x + 4\) is all real numbers.

Answer:

All real numbers