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?

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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.