Problem

For \(f(x)~=~|x~-~4|~+~1\), state the turning point, restrict the function to each monotonic branch, write \(\text{both}~\text{inverse}~\text{functions}\), and give the branch ranges and inverse domains.

Prompt coordinate graph of f(x)=|x−4|+1, a V-shaped graph with turning point (4,1), shown without branch restrictions or inverse reflections.

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Turning point: (4,1). Branches: increasing on x≤4 with inverse x+3; decreasing on x≥4 with inverse 5−x. Both branch ranges and inverse domains are [1,∞), and each restriction is one-to-one.

Turning point: (4,1). Branches: increasing on x≥4 with inverse x+3; decreasing on x≤4 with inverse 5−x. Both branch ranges and inverse domains are [1,∞), and each restriction is one-to-one.

Turning point: (4,1). Branches: increasing and decreasing on all real x, with inverses x+3 and 5−x. The range and inverse domain are [1,∞), and each unrestricted branch is one-to-one.

Turning point: (4,1). Branches: increasing on x≥4 with inverse x−3; decreasing on x≤4 with inverse 5−x. Both branch ranges and inverse domains are [1,∞), and each restriction is one-to-one.

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For \(f(x)~=~|x~-~4|~+~1\), state the turning point, restrict the function to each monotonic branch, write \(\text{both}~\text{inverse}~\text{functions}\), and give the branch ranges and inverse domains.

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