Let f : ℝ →ℝ defined by f(x)=a x+b, where a, b ∈ℝ and a ≠0 . Show that f is surjective; that is,

step 1 So we need to show that this is subjective, right? So we need to show that f of x is equal to c.

Let f : ℝ →ℝ defined by f(x)=a x+b, where a, b ∈ℝ and a ≠0 ....

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Key Concepts

Linear Function

A linear function is a function of the form f(x) = ax + b, where a and b are constants and a ? 0. This form represents a straight line when graphed on the Cartesian plane, and its properties, such as slope and intercept, are fundamental in various areas of mathematics. Understanding linear functions is crucial, as they provide basic examples of functions that are both continuous and bijective under appropriate restrictions.

Surjectivity

Surjectivity, or being 'onto', is a property of a function that ensures every element in the codomain has at least one pre-image in the domain. Demonstrating that a linear function is surjective involves showing that for any element c in the codomain, there exists an x in the domain such that f(x) = c. This concept is essential in understanding how functions cover their output spaces.

Algebraic Manipulation for Function Inversion

Algebraic manipulation is used to invert a function by solving for the independent variable x in terms of the dependent variable c. For linear functions, this is straightforward since the equation ax + b = c can be rearranged to x = (c - b)/a. This process highlights the concept of an inverse function and is a fundamental tool in solving equations and analyzing function properties.

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