Find the range of the given function.
f, of, x, equals, vertical bar, 4, x, minus, 20, vertical bar, plus, 5
f(x)=∣4x−20∣+5

To find the range of the function \( f(x) = |4x - 20| + 5 \), let's break it down step by step.

### Step 1: Understanding the Absolute Value
The function \( f(x) \) has an absolute value term, \( |4x - 20| \). The absolute value of any expression is always non-negative, i.e., \( |4x - 20| \geq 0 \) for all values of \( x \).

Therefore:
\[
|4x - 20| + 5 \geq 0 + 5 = 5
\]
This implies that the minimum value of \( f(x) \) is 5.

### Step 2: Minimum Value of \( f(x) \)
To find the specific point at which \( f(x) \) takes this minimum value, we solve for \( 4x - 20 = 0 \):
\[
4x = 20 \implies x = 5
\]
At \( x = 5 \), we have:
\[
f(5) = |4(5) - 20| + 5 = |20 - 20| + 5 = 0 + 5 = 5
\]
Thus, the minimum value of \( f(x) \) is 5.

### Step 3: Range of the Function
Since \( |4x - 20| \) can take any non-negative value and grows larger as \( |x| \) increases, the term \( |4x - 20| + 5 \) can take values from 5 up to infinity. Therefore, the range of the function is:
\[
\boxed{[5, \infty)}
\]

### Conclusion
The range of \( f(x) = |4x - 20| + 5 \) is \( [5, \infty) \).

Would you like any further details or clarifications?

---

Here are five related questions to explore further:

1. What is the domain of the function \( f(x) = |4x - 20| + 5 \)?
2. How does the graph of the function \( f(x) = |4x - 20| + 5 \) look?
3. What is the range of \( f(x) = |4x + 20| + 5 \), a similar function?
4. Can we rewrite the absolute value function \( f(x) = |4x - 20| + 5 \) as a piecewise function?
5. What happens to the range if we modify the function to \( f(x) = 2|4x - 20| + 5 \)?

**Tip:** Absolute value functions always have their minimum at the point where the expression inside the absolute value equals zero.

Find the range of the given function f(x) = |4x - 20| + 5

Understand how to find the range of the absolute value function f(x) = |4x - 20| + 5 with a step-by-step breakdown. This problem explores the properties of absolute value functions and their impact on determining the range. Ideal for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation