One-to-one functions

1. Is $f\left(x\right)=x^2$ one-to-one on the real numbers? Justify your answer.

   2 marks
2. Show $f\left(x\right)=3x-7$ is one-to-one on the real numbers.

   3 marks
3. Is $f\left(x\right)=\frac{1}{x}$, $x\ne 0$, one-to-one?

   3 marks
4. Find two distinct inputs with the same output for $f\left(x\right)=x^3-3x$.

   3 marks
5. Give two maximal intervals on which $f\left(x\right)={\left(x-2\right)}^2$ is one-to-one.

   3 marks
6. Explain why $f\left(x\right)=2^x$ has at most one input for each output.

   2 marks

Equivalent correct methods earn the corresponding marks. Include units where asked.

1. Is $f\left(x\right)=x^2$ one-to-one on the real numbers? Justify your answer.

   2 marks

   Use $f\left(2\right)=f\left(-2\right)=4$ even though the inputs differ.

   No.
2. Show $f\left(x\right)=3x-7$ is one-to-one on the real numbers.

   3 marks

   If $3a-7=3b-7$, then $3a=3b$ and $a=b$.

   It is one-to-one.
3. Is $f\left(x\right)=\frac{1}{x}$, $x\ne 0$, one-to-one?

   3 marks

   If $\frac{1}{a}=\frac{1}{b}$ for nonzero $a,b$, multiplying by $ab$ gives $a=b$.

   Its separated branches do not share any y-value.

   Yes.
4. Find two distinct inputs with the same output for $f\left(x\right)=x^3-3x$.

   3 marks

   $x^3-3x=x\left(x^2-3\right)$.

   Use two different roots.

   $0$ and $\sqrt{3}$ both give $0$.
5. Give two maximal intervals on which $f\left(x\right)={\left(x-2\right)}^2$ is one-to-one.

   3 marks

   The graph is strictly decreasing on the first interval and increasing on the second.

   Each restriction passes the horizontal-line test.

   $\left(-\infty ,2\right]$ and $\left[2,\infty \right)$.
6. Explain why $f\left(x\right)=2^x$ has at most one input for each output.

   2 marks

   For $a<b$, raising a base greater than 1 to these powers gives $2^a<2^b$.

   A horizontal line meets its graph at most once.

   It is strictly increasing.