Determine if the function is linear, constant, or neither.

Part 1 of 4

linear
constant
neither linear nor constant

Part 2 of 4
linear
Check
Save For Late

Ask by Allan Cook. in the United States

Jan 23,2025

Question

Determine if the function is linear, constant, or neither. Part 1 of 4 $$ h(x)=-\frac{4}{5 x}-1 $$ linear constant neither linear nor constant Part 2 of 4 $$ \square $$ linear Check Save For Late

UpStudy AI · Accepted Answer

To determine whether the function $$ h(x) = -\frac{4}{5x} - 1 $$ is **linear**, **constant**, or **neither**, let's analyze its form.

### **Definition of Terms:**

1. **Linear Function:**
   - A function is **linear** if it can be expressed in the form $$ h(x) = mx + b $$, where $$ m $$ and $$ b $$ are constants.
   - The graph of a linear function is a straight line.

2. **Constant Function:**
   - A function is **constant** if it does not depend on $$ x $$ and can be written as $$ h(x) = c $$, where $$ c $$ is a constant.
   - The graph of a constant function is a horizontal line.

### **Analyzing $$ h(x) = -\frac{4}{5x} - 1 $$:**

1. **Presence of $$ x $$ in the Denominator:**
   - The term $$ -\frac{4}{5x} $$ involves $$ x $$ in the denominator, making the function a **rational function**.
   - Rational functions are generally **not linear** because they include terms like $$ \frac{1}{x} $$, $$ \frac{x}{x} $$ (which simplifies to a constant), or higher powers of $$ x $$ in the denominator.

2. **Dependence on $$ x $$:**
   - The function depends on $$ x $$ due to the $$ \frac{1}{x} $$ term, so it is **not constant**.

### **Conclusion:**

Since $$ h(x) $$ includes a term with $$ \frac{1}{x} $$ and cannot be expressed in the form $$ mx + b $$, it is **neither linear nor constant**.

**Answer:**  
The function $$ h(x) = -\frac{4}{5x} - 1 $$ is **neither linear nor constant**.
The function $$ h(x) = -\frac{4}{5x} - 1 $$ is **neither linear nor constant**.

Determine if the function is linear, constant, or neither.

Part 1 of 4

linear
constant
neither linear nor constant

Part 2 of 4
linear
Check
Save For Late

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Determine if the function is linear, constant, or neither. Part 1 of 4 linear constant neither linear nor constant Part 2 of 4 linear Check Save For Late