MATH1202 Chap.5 Inverse, Exponential and Logarithmic Functions
Inverse, Exponential and Logarithmic Functions
This chapter treats inverse functions as reversed input-output processes and uses composition as the decisive check. Exponential and logarithmic functions form an inverse pair, so their equations, domains, ranges and graphs are linked. Log laws are derived from index laws, and exponential models are interpreted through multipliers, asymptotes and time units.
The reflection of a one-to-one graph in the line y equals x provides a geometric check on swapped domain and range. Equation solving keeps positivity conditions visible before logarithms are combined. Model parameters are read as an initial value, a multiplier or rate, and a time scale rather than handled as detached symbols.
What this chapter covers
- 01
One-to-one inverse condition
- 02
Algebraic inverse construction
- 03
Composition verification
- 04
Exponential and logarithmic equivalence
- 05
Log product, quotient and power laws
- 06
Growth, decay and asymptotes
Worked example · free
Solve a logarithmic equation
- +1State positive-argument conditions and combine them.
- +1Combine the logarithms into one product.
- +1Convert to exponential form and solve the quadratic.
- +1Reject the candidate outside the original domain.
Key terms
- One-to-One
- A function property where distinct inputs produce distinct outputs.
- Inverse Function
- A function reversing the action of a one-to-one function.
- Logarithm
- The exponent required to produce a positive number from a valid base.
- Asymptote
- A limiting line controlling the graph near a boundary or over time.
Inverse, Exponential and Logarithmic Functions FAQ
Why does a parabola need a restricted domain for an inverse?
An unrestricted parabola sends two symmetric inputs to the same output, so reversing it would assign two outputs to one input. Restricting to one side makes the rule one-to-one and the inverse relation a function.
Why can logarithm equations create invalid roots?
Combining logs and solving the resulting algebraic equation can produce candidates that make an original logarithm argument zero or negative. State positivity conditions first and test every candidate in the original equation.
When should I use logarithms in an exponential model?
Use logarithms when the unknown appears in an exponent, often a time. Keep the model units visible, solve with exact logarithms, and round only in the final contextual statement.
How do I verify an inverse rule?
Compose the proposed inverse with the original function in both orders. Each composition should return its input on the appropriate domain. Also compare domain and range after swapping them and check that corresponding graph points reverse their coordinates. A single algebraic rearrangement is not enough if a domain restriction was introduced or lost.
What conditions do logarithm laws require?
Every logarithm argument must be positive, and the base must be positive and not equal to one. Product, quotient and power laws do not override these conditions. State them before combining expressions, because the compact logarithm may appear defined at a value that made one of the original separate logarithms invalid.
How do growth and decay differ in a model?
A positive initial amount multiplied by a factor greater than one grows each period; a factor strictly between zero and one decays. In a continuous exponential form, the sign of the rate parameter plays the same role. Identify the time unit, compare values one period apart and locate any horizontal asymptote before interpreting long-run behaviour.
Why can rounding a model time upward matter?
A continuous solution may occur between available observation or payment times. If the question asks when a threshold has first been reached at whole periods, test the adjacent integers and choose the first one satisfying the original inequality. Ordinary nearest-number rounding can choose a time at which the requirement is still unmet.
How are exponential and logarithmic graphs linked?
With the same valid base, the logarithmic graph is the inverse of the exponential graph. Reflection in the line y equals x swaps their coordinates, domains, ranges and horizontal or vertical asymptotes. Matching points such as an exponential output of one and a logarithmic input of one provide anchors. This relationship also checks equation conversions and prevents assigning impossible logarithm inputs.
Exam move
Pair every inverse calculation with a composition check and a domain-range swap. Sketch a function, its inverse and the line y equals x on the same axes to connect the algebra with reflection. Convert repeatedly between exponential and logarithmic forms until the equivalence is automatic, using exact values before calculator approximations.
For logarithmic equations, write the domain first, combine using one law at a time, solve, and reject invalid candidates in the original equation. Build a short table of common exponential values and their matching logarithms. For models, identify the initial amount, multiplier or continuous rate, asymptote and time unit before calculating a threshold time.
Compare adjacent time values whenever the context is discrete, and explain the final rounding direction in words.
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