Question
Find the vertex of the given function. \( f(x)=|x-5|+10 \) The vertex is at 5 Complete What are the domain and range of the function? domain: \( (-\infty, \infty) \); range: \( f(x) \geq-5 \) domain: \( (-\infty, \infty) \); range: \( f(x) \geq 10 \) domain: \( x \geq 5 \); range: \( f(x) \geq 10 \) domain: \( f(x) \geq 5 \); range: \( (-\infty, \infty) \) DONE
Ask by Donnelly Byrd. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The vertex is at (5, 10).
Domain: All real numbers
Range: f(x) ≥ 10
Solution
The given function is f(x) = |x – 5| + 10.
Step 1: Find the vertex.
An absolute value function of the form f(x) = |x – h| + k has its vertex at (h, k).
For f(x) = |x – 5| + 10, the vertex is (5, 10).
Step 2: Determine the domain.
The absolute value function is defined for all real numbers.
Domain: (–∞, ∞).
Step 3: Determine the range.
Since |x – 5| is always nonnegative, the smallest value f(x) can have is when |x – 5| = 0, which makes f(5) = 10.
Range: f(x) ≥ 10 (or [10, ∞)).
Among the answer options provided, the correct one is:
"domain: (-∞, ∞); range: f(x) ≥ 10".
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Extra Insights
The function \( f(x) = |x-5| + 10 \) is a V-shaped graph that opens upwards. The vertex, which is the lowest point of this function, occurs at the point (5, 10). This means that for any x-value less than 5, the function outputs values greater than 10, and it only increases from there. In terms of the provided options, the correct answer for the domain and range is domain: \( (-\infty, \infty) \); range: \( f(x) \geq 10 \). The function is defined for all x-values, while the function values (range) start from 10 and go upwards, aligning perfectly with the V-shape's lowest point at the vertex.
