Practice Problems for 1/13
- For a function \(f:A \rightarrow B\), \(A\) is called the domain and \(B\) is called the co-domain.
- True or False: The range of a function must be equal to its co-domain. (Write one sentence justifying your response.)
- False, the range must be contained in the codomain.
- True or False: The map \(f: \mathbb{R}\rightarrow \mathbb{R}\) by \(f(x)=x^2-1\) is a function. (Write one sentence justifying your response.)
- True- each real number is sent to one unique real number.
- True or False: The map \(f: \mathbb{R}\rightarrow [0,\infty)\) by \(f(x)=x^2-1\) is a function. (Write one sentence justifying your response.)
- False, since the codomain has been restricted, there is no where for the elements in (-1,1) to be sent.
- Write a step by step procedure for how one could check whether or not a function \(f\) is an odd function.
- Compute \(f(-x)\).
- Compare \(f(x)\) and \(f(-x)\).
- If \(f(-x)=-f(x)\) conclude the fuction is odd. If not, conclude the function is not odd.
Is it possible for a function to be both even and odd? Explain.
- If a function is even then \(f(-x)=f(x)\). If it were both even and odd we would require that \(f(x)=-f(x)\) so that both criteria are satified. This is only possible if we consider the trivial constant function \(f(x)=0\).
- Suppose you would like to graph both \(f(x)=x^3\) and \(g(x)=\frac{1}{2}(4x)^3-1\). Make a bullet point list of each change (i.e. vertical/horizontal shifts and/or scales) you would need to carry out to get from \(f\) to \(g\).
- Compress (i.e. scale to make smaller) horizontally by a factor of 4.
- Compress vertically by a factor of 2.
- Shift vertically by -1.