Question

The graph of f(x) = |x| is translated 6 units to the right and 2 units up to form a new function. Which statement about the range of both functions is true?

The range is the same for both functions: {y | y is a real number}.
The range is the same for both functions: {y | y > 0}.
The range changes from {y | y > 0} to {y | y > 2}.
The range changes from {y | y > 0} to {y | y > 6}.

Answer 1

To determine how the range of the function \( f(x) = |x| \) changes after the translation, we first analyze the original function.

**Original function:**
- The range of \( f(x) = |x| \) is \( \{ y | y \geq 0 \} \) because the absolute value function produces non-negative outputs.

**Translation:**
- Translating the graph 6 units to the right involves replacing \( x \) with \( x - 6 \), resulting in the function \( f(x) = |x - 6| \).
- Translating the graph 2 units up means adding 2 to the function, creating the new function \( g(x) = |x - 6| + 2 \).

Now we analyze the range of the new function \( g(x) \):
- The minimum value of \( |x - 6| \) is 0, which occurs when \( x = 6 \).
- Therefore, the minimum value of \( g(x) = |x - 6| + 2 \) is \( 0 + 2 = 2 \).
- As \( |x - 6| \) can take increasing values from 0 to \( \infty \), \( g(x) \) can take values from 2 to \( \infty \).
- Thus, the range of \( g(x) \) is \( \{ y | y \geq 2 \} \).

**Comparing the ranges:**
- The range of the original function \( f(x) = |x| \) is \( \{ y | y \geq 0 \} \).
- The range of the new function \( g(x) = |x - 6| + 2 \) is \( \{ y | y \geq 2 \} \).

**Conclusion:**
The correct statement is that the range changes from \( \{ y | y \geq 0 \} \) to \( \{ y | y \geq 2 \} \).

Therefore, the correct answer is:
**The range changes from {y | y > 0} to {y | y > 2}.**