To find the range of the function [tex]\( f(x) = \frac{2x - 1}{x + 3} \)[/tex], we need to analyze its behavior and identify any values that the function cannot take.
### Step-by-Step Solution
1. Identify the Domain:
The function [tex]\( f(x) \)[/tex] will be undefined where the denominator is zero:
[tex]\[
x + 3 = 0 \implies x = -3
\][/tex]
Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers except [tex]\( x = -3 \)[/tex].
2. Explore the Function Behavior:
To identify the range, we need to solve for [tex]\( y \)[/tex] in the equation [tex]\( y = \frac{2x - 1}{x + 3} \)[/tex] in terms of [tex]\( x \)[/tex].
3. Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = \frac{2x - 1}{x + 3}
\][/tex]
Multiply both sides by [tex]\( x + 3 \)[/tex]:
[tex]\[
y(x + 3) = 2x - 1
\][/tex]
Expand and rearrange:
[tex]\[
yx + 3y = 2x - 1 \implies yx - 2x = -3y - 1 \implies x(y - 2) = -3y - 1
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-3y - 1}{y - 2}
\][/tex]
4. Find the Undefined Value for [tex]\( y \)[/tex]:
The expression for [tex]\( x \)[/tex] will be undefined if the denominator is zero:
[tex]\[
y - 2 = 0 \implies y = 2
\][/tex]
This means that [tex]\( f(x) \)[/tex] cannot take the value [tex]\( y = 2 \)[/tex].
5. Verify with the Function Behavior:
As [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] (where the function is undefined), the function tends to the limits:
[tex]\[
\lim_{x \to -3^-} \frac{2x - 1}{x + 3} \rightarrow -\infty
\][/tex]
[tex]\[
\lim_{x \to -3^+} \frac{2x - 1}{x + 3} \rightarrow \infty
\][/tex]
All other values of [tex]\( y \)[/tex] are attainable by [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] varies over the real numbers, except for [tex]\( y = 2 \)[/tex].
### Conclusion
The function [tex]\( f(x) = \frac{2x - 1}{x + 3} \)[/tex] can take any real value except [tex]\( 2 \)[/tex]. Hence, the range of the function is:
[tex]\[
\boxed{\mathbb{R} \backslash \{2\}}
\][/tex]
