The functions defined have domain [-1,1] and has values in...

The functions defined have domain $[-1,1]$ and has values in $[-12,8]$ (a) $2 x^{3}-3 x^{2}-12 x+1$ (b) $\sqrt{\left(1-x^{2}\right)\left(1+2 x^{2}\right)}$ (c) $\frac{1-x+x^{2}}{1+x+x^{2}}$ (d) $x^{3}-3 x^{2}+6 x-2$ (p) is one-one but not onto (q) is one-one and onto (r) is ncither onc-one nor onto (s) is not onc-one but onto

The functions defined have domain $[-1,1]$ and has values in $[-12,8]$
(a) $2 x^{3}-3 x^{2}-12 x+1$
(b) $\sqrt{\left(1-x^{2}\right)\left(1+2 x^{2}\right)}$
(c) $\frac{1-x+x^{2}}{1+x+x^{2}}$
(d) $x^{3}-3 x^{2}+6 x-2$

(p) is one-one but not onto
(q) is one-one and onto
(r) is ncither onc-one nor onto
(s) is not onc-one but onto

Key Concepts

Radical Functions

Radical functions involve roots, such as square roots, and necessitate careful consideration of the domain since the expression inside the radical (the radicand) must satisfy conditions (like non-negativity for even-indexed roots) to produce real number outputs.

Polynomials

Polynomials are algebraic functions consisting of terms with nonnegative integer exponents. Their continuity and smooth behavior over closed intervals make them suitable for studying properties like injectivity and determining range, especially when domain restrictions are applied.

Surjective Function (Onto)

A surjective (onto) function is one in which every element of the codomain has at least one preimage in the domain. This means the function’s output covers the entire codomain, and analysis typically requires ensuring that no values within the codomain are excluded by the mapping process.

Rational Functions

Rational functions, which are ratios of two polynomials, often require special attention to ensure that the denominator does not become zero. Their behavior can include asymptotes, discontinuities, or non-obvious restrictions on their range, making a detailed analysis necessary to assess properties like injectivity or surjectivity.

Range

The range of a function is the set of all possible outputs it can produce given its domain. Determining the range involves understanding the behavior of the function over its entire domain and includes the consideration of any restrictions caused by the function’s algebraic form.

Domain

The domain of a function is the set of all possible inputs for which the function is defined. When analyzing any function, it is important to check the domain to ensure that all operations (such as square roots or division) are valid for the chosen inputs.

Injective Function (One-One)

An injective (one-one) function is characterized by the property that each element of the function’s domain maps to a unique element in the codomain. This means no two distinct inputs produce the same output, a property that is often verified by checking if the function is strictly increasing or decreasing over its domain.

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