Problem

For \(f(x)~=~x^2\), 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)=x², an upward-opening parabola with turning point (0,0), shown without branch restrictions or inverse reflections.

Big Picture

What this problem is really about

A parabola fails to be one-to-one because its two sides repeat outputs, so the turning point is the natural place to split it into maximal monotonic branches. Invert each branch separately, using its sign restriction to decide the square-root sign. The shared branch range becomes each inverse domain; a single plus-or-minus expression would not define one inverse function.

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Turning point: (0,0). Branches: increasing on x≥0 with inverse √x; decreasing on x≤0 with inverse −√x. Both branch ranges and inverse domains are [0,∞), and each restriction is one-to-one.

Turning point: (0,0). Branches: increasing and decreasing on all real x, with inverses √x and −√x. The range and inverse domain are [0,∞), and each unrestricted branch is one-to-one.

Turning point: (0,0). Branches: increasing on x≤0 with inverse √x; decreasing on x≥0 with inverse −√x. Both branch ranges and inverse domains are [0,∞), and each restriction is one-to-one.

Turning point: (0,0). Branches: increasing on x≥0 with inverse x²; decreasing on x≤0 with inverse −x². Both branch ranges and inverse domains are [0,∞), and each restriction is one-to-one.

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For \(f(x)~=~x^2\), 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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