Functions Practice Hard Test

Practice function notation, transformations, inverse functions, exponential functions, and logarithmic functions with free online tests from easy to harder levels.

Available tests

Choose a Functions test
Test 1
30 questions
Elementary functions
Hard Level
Test 2
30 questions
Function transformation
Hard Level
Test 3
30 questions
Inverse functions
Hard Level
Test 4
30 questions
Exponential functions
Hard Level
Test 5
30 questions
Logarithmic functions
Hard Level

Free Functions practice tests

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Topic-based practice

Practice different functions skills

x → f(x) → graph → inverse

A function is more than a formula — it connects inputs, outputs, graphs, and models

The five practice tests follow that connection from basic function notation to transformations, inverse relationships, exponential growth, and logarithms. Instead of treating these as separate chapters, use the sequence to see how each idea builds on the previous one.

Start here · Test 01 30 questions

Elementary Functions

Begin with the language used everywhere else on this page: function notation, inputs and outputs, domain, range, evaluating formulas, and recognizing common function types. These skills make later graph and inverse problems much easier to read.

f(x) f(3) domain range
Change the graph · Test 02 30 questions

Function Transformations

Learn to read changes in a formula as movements or changes in a graph. Practice horizontal and vertical shifts, reflections, stretches, and compressions while keeping track of which transformations happen inside the function and which happen outside it.

f(x) + k f(x - h) -f(x) f(-x)
Reverse the relationship · Test 03 30 questions

Inverse Functions

An inverse reverses what the original function does. Practice switching inputs and outputs, finding inverse rules, checking compositions, and recognizing when a function must be restricted before an inverse can be defined as a function.

f-1(x) f(f-1(x)) = x one-to-one
Model repeated change · Test 04 30 questions

Exponential Functions

Move from additive change to multiplicative change. Practice exponential growth and decay, evaluating exponential expressions, comparing models, and interpreting how the base affects the behavior of the graph.

y = abx b > 1 0 < b < 1 growth / decay
Undo the exponent · Test 05 30 questions

Logarithmic Functions

Finish the sequence by connecting logarithms to inverse functions and exponentials. Practice converting between exponential and logarithmic forms, using log properties, solving equations, and interpreting logarithmic graphs.

logbx by = x logbx = y
Common traps

Three mistakes worth catching early

Function questions often go wrong because of notation or interpretation rather than difficult algebra.

Read inside transformations carefully

In an expression such as f(x − 3), the graph moves to the right, not to the left. Changes inside the function often behave opposite to the sign you first see.

Do not ignore the domain

A formula may be algebraically valid but still exclude inputs because of division by zero, even roots, or logarithms of nonpositive values.

An inverse is not a reciprocal

f-1(x) means the inverse function, not 1/f(x). A true inverse reverses the input-output relationship of the original function.

Use the tests as a sequence when a topic feels weak

If inverse, exponential, or logarithmic questions feel harder than expected, return to the earlier tests first. Problems later in the sequence often assume that function notation, transformations, and domain restrictions are already comfortable.

Questions students run into

Function notation, graphs, inverses, and logarithms

Short explanations for ideas that commonly cause confusion while working through function problems.

What does f(x) actually mean?

The notation f(x) represents the output of the function f when the input is x. If f(x) = 2x + 3, then f(4) means replace x with 4, giving 2(4) + 3 = 11.

What is the difference between domain and range?

The domain is the set of allowed input values. The range is the set of output values the function can produce. Restrictions in a formula often determine the domain, while the behavior of the function determines the range.

Why does f(x − 2) shift a graph to the right?

The transformed function produces the original output when x − 2 reaches the old input value. That happens two units later on the x-axis, so every point of the graph is shifted two units to the right.

How do I know whether a function has an inverse?

A function has an inverse function on a domain when each output corresponds to only one input. Graphically, this is connected to the horizontal line test. Some functions can become one-to-one after restricting their domain.

What is the main difference between exponential and linear growth?

Linear growth adds the same amount over equal intervals. Exponential growth multiplies by the same factor over equal intervals. Because the amount being multiplied keeps changing, exponential models can eventually grow much faster than linear ones.

How are logarithmic and exponential functions related?

They are inverse functions. The statement by = x is equivalent to logb(x) = y. Exponential notation asks for an output after applying a power, while logarithmic notation asks which exponent produces a given output.

Why must the input of a logarithm be positive?

For the real logarithmic functions normally used in algebra and precalculus, a positive base raised to a real power always produces a positive result. Therefore the inverse logarithmic function accepts only positive inputs.

What should I review if I keep making mistakes with functions?

Check whether the mistake comes from function notation, algebra, domain restrictions, graph transformations, or misunderstanding an inverse. Identifying the type of error is more useful than simply repeating the same calculation.