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Function Composition to Domain - Integer over Root of Quadratic (Real Roots) to Domain Definition

Level 1

This math topic focuses on the composition of functions and determining their domains, specifically involving operations with integers over the root of quadratic expressions, ensuring these quadratics have real roots. Students must understand how to determine the domain from the composition of an outer function in terms of the square root applied to a quadratic inner function. They are provided multiple choice answers for each problem, requiring deep analysis of quadratic expressions and restrictions imposed by the function's domain, such as evaluating where the input for the square root is non-negative and real. This is introduced as a Level 1 introductory topic to function composition and inversion.

Work on practice problems directly here, or download the printable pdf worksheet to practice offline.

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Function Composition to Domain - Integer over Root of Quadratic (Real Roots) to Domain Definition Worksheet

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Function Composition to Domain - Integer over Root of Quadratic (Real Roots) to Domain Definition
1
A LaTex expression showing \begin{align*}f(x)&=3 over square root of x \\\\g(x)&=1x to the power of 2 -0x-9\\\\f(g(x)) &\rightarrow \text{Domain}?\end{align*}\\
Which set describes the domain of this function composition?
a A LaTex expression showing \{X \in \mathbb{{R}} \vert -3 < X < 3\text{{ or }}-3 < X < 3\}
b A LaTex expression showing \{X \in \mathbb{{R}} \vert X < -3\text{{ or }}3 < X\}
2
A LaTex expression showing \begin{align*}f(x)&=5 over square root of x \\\\g(x)&=2x to the power of 2 -10x-0\\\\f(g(x)) &\rightarrow \text{Domain}?\end{align*}\\
Which set describes the domain of this function composition?
a A LaTex expression showing \{X \in \mathbb{{R}} \vert X < 0\text{{ or }}-5 < X < 5\}
b A LaTex expression showing \{X \in \mathbb{{R}} \vert X < 0\text{{ or }}5 < X\}
3
A LaTex expression showing \begin{align*}f(x)&=3 over square root of x \\\\g(x)&=1x to the power of 2 -5x-14\\\\f(g(x)) &\rightarrow \text{Domain}?\end{align*}\\
Which set describes the domain of this function composition?
a A LaTex expression showing \{X \in \mathbb{{R}} \vert X < -2\text{{ or }}7 < X\}
b A LaTex expression showing \{X \in \mathbb{{R}} \vert X < -2\text{{ or }}-7 < X < 7\}
4
A LaTex expression showing \begin{align*}f(x)&=-3 over square root of x \\\\g(x)&=2x to the power of 2 -6x-20\\\\f(g(x)) &\rightarrow \text{Domain}?\end{align*}\\
Which set describes the domain of this function composition?
a A LaTex expression showing \{X \in \mathbb{{R}} \vert X < -2\text{{ or }}5 < X\}
b A LaTex expression showing \{X \in \mathbb{{R}} \vert -2 < X < 2\text{{ or }}-5 < X < 5\}
5
A LaTex expression showing \begin{align*}f(x)&=-3 over square root of x \\\\g(x)&=-3x to the power of 2 -9x-6\\\\f(g(x)) &\rightarrow \text{Domain}?\end{align*}\\
Which set describes the domain of this function composition?
a A LaTex expression showing \{X \in \mathbb{{R}} \vert -2 < X < -1\}
b A LaTex expression showing \{X \in \mathbb{{R}} \vert -2 \le X < -1\}
6
A LaTex expression showing \begin{align*}f(x)&=2 over square root of x \\\\g(x)&=-2x to the power of 2 -10x+12\\\\f(g(x)) &\rightarrow \text{Domain}?\end{align*}\\
Which set describes the domain of this function composition?
a A LaTex expression showing \{X \in \mathbb{{R}} \vert -6 < X < 1\}
b A LaTex expression showing \{X \in \mathbb{{R}} \vert -6 \le X \le 1\}
7
A LaTex expression showing \begin{align*}f(x)&=2 over square root of x \\\\g(x)&=2x to the power of 2 -6x-0\\\\f(g(x)) &\rightarrow \text{Domain}?\end{align*}\\
Which set describes the domain of this function composition?
a A LaTex expression showing \{X \in \mathbb{{R}} \vert X < 0\text{{ or }}3 < X\}
b A LaTex expression showing \{X \in \mathbb{{R}} \vert X < 0\text{{ or }}0 \le X \le 3\}
8
A LaTex expression showing \begin{align*}f(x)&=-1 over square root of x \\\\g(x)&=1x to the power of 2 +1x-12\\\\f(g(x)) &\rightarrow \text{Domain}?\end{align*}\\
Which set describes the domain of this function composition?
a A LaTex expression showing \{X \in \mathbb{{R}} \vert X < -4\text{{ or }}3 < X\}
b A LaTex expression showing \{X \in \mathbb{{R}} \vert X < -4\text{{ or }}-3 < X < 3\}