Asked by htht
Which of the pairs of functions and their inverses will always have a domain and range of all real numbers?
a pair of linear functions with defined slopes
a cubic function and a cube root function
a quadratic function and a square root function
a logarithmic function and an exponential function
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GPT-5 mini
AI
Correct: the linear pair and the cubic/cube-root pair.
- A nonhorizontal linear function f(x)=mx+b (m ≠ 0) has domain (−∞,∞) and range (−∞,∞); its inverse is also linear with domain and range all reals.
- The cubic function f(x)=x^3 has domain and range (−∞,∞), and its inverse f⁻¹(x)=∛x also has domain and range (−∞,∞). (Note: a general cubic polynomial need not be one-to-one, but the standard cube function x^3 and its cube-root inverse do.)
The quadratic/square-root and exponential/logarithm pairs do not: quadratics typically have restricted ranges (e.g. [k,∞)), and exponentials/logarithms have range or domain (0,∞), not all real numbers.
- A nonhorizontal linear function f(x)=mx+b (m ≠ 0) has domain (−∞,∞) and range (−∞,∞); its inverse is also linear with domain and range all reals.
- The cubic function f(x)=x^3 has domain and range (−∞,∞), and its inverse f⁻¹(x)=∛x also has domain and range (−∞,∞). (Note: a general cubic polynomial need not be one-to-one, but the standard cube function x^3 and its cube-root inverse do.)
The quadratic/square-root and exponential/logarithm pairs do not: quadratics typically have restricted ranges (e.g. [k,∞)), and exponentials/logarithms have range or domain (0,∞), not all real numbers.
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