What Determines the Domain for Inverses of Quadratic Equations?

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SUMMARY

The discussion clarifies the determination of domains for the inverses of quadratic equations, specifically addressing the functions x² - 4 and 2x² + 3. Both functions can have multiple valid domains, including x ≥ 0 and x ≤ 0, which allow for their inverses to exist. The largest domain is defined as all points on one side of the vertex, which for both functions is at x = 0. Thus, the largest domains for these quadratics are x ≥ 0 and x ≤ 0.

PREREQUISITES
  • Understanding of quadratic functions and their properties
  • Knowledge of the concept of function inverses
  • Familiarity with graphing parabolas and identifying vertices
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of quadratic functions and their graphs
  • Learn about the concept of one-to-one functions and their inverses
  • Explore methods for finding the vertex of a parabola
  • Investigate the implications of domain restrictions on function behavior
USEFUL FOR

Students and educators in mathematics, particularly those focusing on algebra and quadratic functions, as well as anyone seeking to understand the conditions for the existence of function inverses.

thornluke
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Hi guys,

I'm really confused in finding the domain of quadratic equations.

For example: when finding a suitable domain so that an inverse exists, why is the domain of x2-4 x>0
whilst,
the domain of 2x2+3 is x≥0

Can the domain of x2-4 be x≥0?

Furthermore, what is the largest domain and how do I know whether or not it is the largest?

Thanks,
Thorn.
 
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thornluke said:
Hi guys,

I'm really confused in finding the domain of quadratic equations.

For example: when finding a suitable domain so that an inverse exists, why is the domain of x2-4 x>0
whilst,
the domain of 2x2+3 is x≥0

Can the domain of x2-4 be x≥0?

Furthermore, what is the largest domain and how do I know whether or not it is the largest?

Thanks,
Thorn.
The assumptions you make for your first question are wrong. The domains of the two functions [itex]x^2- 4[/itex] and [itex]2x^2+ 3[/itex] are not different. There exist an infinite number of domains on which [itex]x^2- 4[/itex] or [itex]2x^2+ 3[/itex] are "one-to-one" and so have inverses. [itex]x\ge 0[/itex] and any subset of that are such domains. So are [itex]x\le 0[/itex] and any subsets of that.

As for finding the largest domain (which is what is normally meant by "domain" without other condition), look at the graph. For these quadratics, the simplest thing to do is to find the vertex of each parabola. For [itex]x^2- 4[/itex], the vertex is (0, -4) and for [itex]2x^2+ 3[/itex], it is (0, 3).

Each parabola "turns back" at its vertex so you have two different x values, on either side of the vertex, with the same y value- the function is not "one to one". As long as your set lies on one side of the vertex, it is a domain on which the function is invertible. A largest possible such domain is all points on one side of the vertex together with the vertex itself. Since the two functions, [itex]x^2- 4[/itex] and [itex]2x^2+ 3[/itex] both have vertex at x= 0, there are two such "largest domains", [itex]x\ge 0[/itex] and [itex]x\le 0[/itex], for both.

Also, since this question has nothing to do with "Abstract and Linear Algebra", I am moving it to "General Math".
 

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