Radical Functions and Inverses

Explore square-root graph, cube-root graph, inverse pair.

Lesson 4 of 53 parts Practice Flashcards

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Square-Root Graph

Objective: Explain square-root graph and use it to solve or interpret a problem.

For y=√(x−h)+k, real inputs satisfy x≥h and the graph starts at (h,k), then increases. The horizontal shift h.

Worked examples

Try itRadical Functions and Inverses — practice 1.1: For f(x)=√(x−5)+3, find f(5).Show answer

Step 1: Substitute x=5: x−5=0. Step 2: The principal square root of zero is zero. Step 3: Add 3 to get 3.

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Square-Root Graph

Objective: Explain square-root graph and use it to solve or interpret a problem.

For y=√(x−h)+k, real inputs satisfy x≥h and the graph starts at (h,k), then increases. The horizontal shift h.

Worked examples

Try itRadical Functions and Inverses — practice 1.1: For f(x)=√(x−5)+3, find f(5).Show answer

Step 1: Substitute x=5: x−5=0. Step 2: The principal square root of zero is zero. Step 3: Add 3 to get 3.

Cube-Root Graph

Objective: Explain cube-root graph and use it to solve or interpret a problem.

A cube-root graph accepts all real inputs and can extend left and right of its central point. Odd-root symmetry differs from a square-root starting edge. Odd powers of negative numbers are negative.

Worked examples

Try itRadical Functions and Inverses — practice 2.1: Find the real cube root of −512.Show answer

Step 1: (−8)³=−512. Step 2: Odd roots preserve the negative sign. Step 3: The answer is −8.

Inverse Pair

Objective: Explain inverse pair and use it to solve or interpret a problem.

Restricted y=x² for x≥0 and y=√x undo each other; the restriction selects one branch of the parabola. Without it, different inputs give the same output.

Worked examples

Try itRadical Functions and Inverses — practice 3.1: If f(x)=√x for x≥0, find f⁻¹(11).Show answer

Step 1: The inverse of the principal square-root function is squaring on x≥0. Step 2: Square the input 11. Step 3: The result is 121.

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