Is the function f(x)=(x-3)/(x^2 +9x -22) injective and surjective?

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∻Solution by Steps∻ ‖ step 1 ⋮ To determine if the function $f(x) = \frac{x-3}{x^2 + 9x - 22}$ is injective, we need to check if $f(x_1) = f(x_2)$ implies $x_1 = x_2$. ‖ ‖ step 2 ⋮ Let $f(x_1) = f(x_2)$. Then, $\frac{x_1 - 3}{x_1^2 + 9x_1 - 22} = \frac{x_2 - 3}{x_2^2 + 9x_2 - 22}$. ‖ ‖ step 3 ⋮ Cross-multiplying gives $(x_1 - 3)(x_2^2 + 9x_2 - 22) = (x_2 - 3)(x_1^2 + 9x_1 - 22)$. ‖ ‖ step 4 ⋮ Expanding and simplifying this equation does not necessarily lead to $x_1 = x_2$, indicating that $f(x)$ is not injective. ‖ ‖ step 5 ⋮ To determine if the function is surjective, we need to check if for every $y \in \mathbb{R}$, there exists an $x \in \mathbb{R}$ such that $f(x) = y$. ‖ ‖ step 6 ⋮ The function $f(x) = \frac{x-3}{x^2 + 9x - 22}$ is not surjective onto $\mathbb{R}$ because there are values of $y$ for which no $x$ satisfies the equation $y = \frac{x-3}{x^2 + 9x - 22}$. ‖ ‖ step 7 ⋮ Specifically, the range of $f(x)$ is restricted by the conditions $169y + 2\sqrt{14} \leq 15$ or $169y \geq 15 + 2\sqrt{14}$. ‖ ∻Answer∻ ⚹ The function $f(x) = \frac{x-3}{x^2 + 9x - 22}$ is neither injective nor surjective. ⚹ ∻Key Concept∻ ⚹Injectivity and Surjectivity of Rational Functions⚹ ∻Explanation∻ ⚹A function is injective if each element of the function's codomain is mapped by at most one element of its domain. It is surjective if every element of the codomain is mapped by at least one element of its domain. The given function fails both criteria.⚹◊Is the function $f(x) = \frac{x-3}{x^2 + 9x - 22}$ invertible?⍭ Generate me a similar question◊

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