# Functions and algebra: Use a variety of techniques to sketch and interpret information for graphs of the inverse of a function

### Subject outcome

Subject outcome 2.2: Use a variety of techniques to sketch and interpret information for graphs of the inverse of a function theorems

### Learning outcomes

- Determine the equations of the inverses of the functions:
- [latex]\scriptsize y=ax+q[/latex]
- [latex]\scriptsize y=a{{x}^{2}}[/latex]
- [latex]\scriptsize y={{a}^{x}},a \gt 0[/latex] ([latex]\scriptsize y={{a}^{x}}[/latex] may be left with [latex]\scriptsize x[/latex] as the subject of the formula. Note: No logarithms required.)

- Sketch the graphs of the inverse of the functions:
- [latex]\scriptsize y=ax+q[/latex]
- [latex]\scriptsize y=a{{x}^{2}}[/latex]
- [latex]\scriptsize y={{a}^{x}},a \gt 0[/latex]

Note: Sketching the graphs using point by point plotting is an option.

- Obtain the equation of any of the following inverse graphs given as a sketch:
- [latex]\scriptsize y=ax+q[/latex]
- [latex]\scriptsize y=a{{x}^{2}}[/latex]
- [latex]\scriptsize y={{a}^{x}},a \gt 0[/latex]

- Identify characteristics as listed below with respect to the following functions and their inverses:

[latex]\scriptsize y=ax+q[/latex]

[latex]\scriptsize y=a{{x}^{2}}[/latex]

[latex]\scriptsize y={{a}^{x}},a \gt 0[/latex]- Domain and range
- Intercepts with axes
- Turning points, minima and maxima
- Asymptotes
- Shape and symmetry
- Functions or non-functions
- Continuous or discontinuous
- Intervals in which a function increases/decreases.

### Unit 1 outcomes

By the end of this unit you will be able to:

- Define an inverse function.
- Find the inverse of [latex]\scriptsize y=ax+q[/latex].
- Sketch the inverse of [latex]\scriptsize y=ax+q[/latex].
- Answer questions about the domain, range, shape, continuity and other characteristics of the inverse graph.

### Unit 2 outcomes

By the end of this unit you will be able to:

- Find the inverse of [latex]\scriptsize y=a{{x}^{2}}[/latex].
- Sketch the inverse of [latex]\scriptsize y=a{{x}^{2}}[/latex].
- Answer questions about the domain, range, shape, continuity and other characteristics of the inverse graph.

### Unit 3 outcomes

By the end of this unit you will be able to:

- Find the inverse of [latex]\scriptsize y={{a}^{x}}[/latex] where [latex]\scriptsize a \gt 0[/latex].
- Sketch the inverse of [latex]\scriptsize y={{a}^{x}}[/latex] where [latex]\scriptsize a \gt 0[/latex].
- Answer questions about the domain, range, shape, continuity and other characteristics of the inverse graph.