Determine Algebraically Whether the Following Function Is One-To-One $f ( x ) = \frac { 5 x ^ { 2 } } { 3 x ^ { 2 } + 6 }$

question 103

Multiple Choice

Determine algebraically whether the following function is one-to-one.
$f ( x ) = \frac { 5 x ^ { 2 } } { 3 x ^ { 2 } + 6 }$ , where $x > 0$

• A)
• \(\begin{aligned}
• \frac { 5 a ^ { 2 } } { 3 a ^ { 2 } + 6 } & = \frac { 5 b ^ { 2 } } { 3 b ^ { 2 } + 6 } \\
• \frac { 5 a ^ { 2 } } { 3 a ^ { 2 } } + \frac { 5 a ^ { 2 } } { 6 } & = \frac { 5 b ^ { 2 } } { 3 b ^ { 2 } } + \frac { 5 b ^ { 2 } } { 6 } \\
• \frac { 5 } { 3 } + \frac { 5 a ^ { 2 } } { 6 } & = \frac { 5 } { 3 } + \frac { 5 b ^ { 2 } } { 6 } \\
• \frac { 30 + 5 a ^ { 2 } } { 18 } & = \frac { 30 + 5 b ^ { 2 } } { 18 } ; \text { not one-to-one } \\
• 30 + 5 a ^ { 2 } & = 30 + 5 b ^ { 2 } \\
• 5 a ^ { 2 } & = 5 b ^ { 2 } \\
• a ^ { 2 } & = b ^ { 2 } \\
• \pm a & = \quad \pm b
• \end{aligned}\)
• B)
• \(\begin{aligned}
• \frac { 5 a ^ { 2 } } { 3 a ^ { 2 } + 6 } & = \frac { 5 b ^ { 2 } } { 3 b ^ { 2 } + 6 } \\
• \frac { 5 } { 3 + 6 } & = \frac { 5 } { 3 + 6 } ; \text { one-to-one } \\
• \frac { 5 } { 6 } & = \frac { 3 } { 6 } \\
• a & = b
• \end{aligned}\)
• C)
• \(\begin{aligned}
• \frac { 5 a ^ { 2 } } { 3 a ^ { 2 } + 6 } & = \frac { 5 b ^ { 2 } } { 3 b ^ { 2 } + 6 } \\
• \frac { 5 a ^ { 2 } } { 3 a ^ { 2 } } & = \frac { 5 b ^ { 2 } } { 3 b ^ { 2 } } ; \text { one-to-one } \\
• \frac { 5 } { 3 } & = \frac { 5 } { 3 } \\
• a & = b
• \end{aligned}\)
• D)
• \(\begin{aligned}
• \frac { 5 a ^ { 2 } } { 3 a ^ { 2 } + 6 } & = \frac { 5 b ^ { 2 } } { 3 b ^ { 2 } + 6 } \\
• \frac { 5 a ^ { 2 } } { 9 a ^ { 2 } } & = \frac { 5 b ^ { 2 } } { 9 b ^ { 2 } } \\
• \frac { 5 a } { 9 } & = \frac { 5 b } { 9 } \\
• 5 a & = 5 b\text { one-to-one } \\
• a & = b
• \end{aligned}\)
• E) \(\frac { 5 a ^ { 2 } } { 3 a ^ { 2 } + 6 } = \frac { 5 b ^ { 2 } } { 3 b ^ { 2 } + 6 }\)
• \(\begin{aligned}
• \frac { 5 a ^ { 2 } } { 3 a ^ { 2 } } + \frac { 5 a ^ { 2 } } { 6 } & = \frac { 5 b ^ { 2 } } { 3 b ^ { 2 } } + \frac { 5 b ^ { 2 } } { 6 } \\
• \frac { 5 } { 3 } + \frac { 5 a ^ { 2 } } { 6 } & = \frac { 5 } { 3 } + \frac { 5 b ^ { 2 } } { 6 } \\
• \frac { 30 + 5 a ^ { 2 } } { 18 } & = \frac { 30 + 5 b ^ { 2 } } { 18 } ; \text { one-to-one } \\
• 30 + 5 a ^ { 2 } & = 30 + 5 b ^ { 2 } \\
• 5 a ^ { 2 } & = 5 b ^ { 2 } \\
• a ^ { 2 } & = b ^ { 2 } \\
• a & = b
• \end{aligned}\)

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