Functions

You will be able to

• Determine the range of a function on a given domain, from the completed square, the end values and any asymptote, and write it in set notation such as f(x) ≥ −3.
• Use the horizontal line test to decide whether a function is one-one or many-one, quoting two x-values with the same image as evidence.
• Restrict the domain of a many-one function so that it becomes one-one.

You will be able to

• Sketch y = |f(x)| from y = f(x) by reflecting the parts below the x-axis in the x-axis, marking where the graph meets the axes.
• Write |f(x)| piecewise and state the range of |f(x)| on a given domain.
• Use a graph to find the number of solutions of |f(x)| = k, solving both f(x) = k and f(x) = −k.

You will be able to

• Form composite functions such as fg(x) = f(g(x)) and evaluate them, knowing that fg and gf are usually different.
• Explain that fg exists only where the range of g lies inside the domain of f, and state the domain of a composite.
• Solve equations involving composite functions, such as ff(x) = 5.

You will be able to

• Find the inverse of a one-one function by writing y = f(x), making x the subject and swapping x and y.
• State the domain of f⁻¹ as the range of f, and the range of f⁻¹ as the domain of f.
• Explain why a many-one function has no inverse, and sketch y = f⁻¹(x) as the reflection of y = f(x) in the line y = x.