Question

Let a, b and c be positive integers. Find the vertex, domain and range of the function f(x)=|ax+b| +c

Answer 1

To analyze the function \( f(x) = |ax + b| + c \), let’s break it down step by step.

### Step 1: Vertex
The expression \( |ax + b| \) has a vertex at the point where the expression inside the absolute value is equal to zero. Solving for \( x \):

\[ ax + b = 0 \]
\[ ax = -b \]
\[ x = -\frac{b}{a} \]

Thus, the x-coordinate of the vertex is \( x = -\frac{b}{a} \).

To find the y-coordinate of the vertex, substitute \( x = -\frac{b}{a} \) into \( f(x) \):

\[
f\left(-\frac{b}{a}\right) = |a(-\frac{b}{a}) + b| + c = | -b + b | + c = |0| + c = c
\]

Therefore, the vertex of the function \( f(x) \) is at:

\[
\left(-\frac{b}{a}, c\right)
\]

### Step 2: Domain
The domain of the function \( f(x) \) is all real numbers since there are no restrictions on \( x \) in the expression \( |ax + b| + c \). Thus, we can express the domain as:

\[
\text{Domain} = (-\infty, \infty)
\]

### Step 3: Range
The term \( |ax + b| \) represents the distance from \( -\frac{b}{a} \) along the y-axis, and since it is always non-negative, we have \( |ax + b| \geq 0 \). Therefore, the minimum value of \( f(x) \) occurs at the vertex:

\[
\text{Minimum value of } f(x) = c
\]

As \( |ax + b| \) can increase indefinitely, the value of \( f(x) \) can also increase indefinitely. Thus, the overall range of \( f(x) \) is:

\[
\text{Range} = [c, \infty)
\]

### Summary
- **Vertex**: \(\left(-\frac{b}{a}, c\right)\)
- **Domain**: \((- \infty, \infty)\)
- **Range**: \([c, \infty)\)