Functions

You will be able to

• Determine the range of a function on a given domain, in correct inequality or interval notation, with the right open or closed ends.
• Explain why gf can only be formed when the range of f lies within the domain of g, and decide whether gf and fg can be formed.
• Determine gf(x) with its domain, and state the inputs that must be excluded with the reason, noting that fg is generally not equal to gf.

You will be able to

• Determine whether a function is one-one on a given domain, using the horizontal-line test and the vertex, and state a restriction that makes it one-one.
• Determine the inverse of a one-one quadratic by completing the square, choosing the root from the domain, and state the domain and range of f⁻¹.
• Describe the graph of f⁻¹ as the reflection of the graph of f in the line y = x, with the domain and range of the two functions swapped.

You will be able to

• Describe fully the transformation given by y = f(x) + a, y = f(x + a), y = af(x) and y = f(ax), using translation, reflection and stretch with the vector, line, direction and scale factor.
• Determine the coordinates of the image of a point under a combination of two steps, and the equation of the image in terms of f.
• Explain why f(x + 2) moves the graph to the left, why f(2x) is a stretch of scale factor 1/2, and why the order of two steps matters.