Mathematics · Composition and inverses
Edited by: Hitoiki Arcade / Hitoiki Learning team · Published: · Editorial policy
Does adding one and then squaring give the same result as squaring and then adding one? With input 2, the first route gives 9 and the second gives 5. When functions are combined, the order in which a value passes through them determines the answer.
A function assigns one output to each allowed input. On paper, write input → intermediate value → output. We will examine composition, reversal and inputs sharing an output, starting with functions defined on all real numbers and later comparing a restricted input domain.
The first exercise defines f(x) = x² and g(x) = x + 1 and asks for f(g(2)). Evaluate the inner g(2) first to get 3, then use that as the input to f: f(3) = 9. The arrows are 2 → 3 under g → 9 under f. The nested parentheses describe this handoff.
Reversing the order gives g(f(2)): 2 → 4 under f → 5 under g. For general inputs, f(g(x)) = (x + 1)² while g(f(x)) = x² + 1, so they are different expressions. Composition also requires the inner output to be an allowed input to the outer function. Both rules here send real numbers to real numbers, so that condition is met.
The second exercise now defines f(x) = 3x − 2. This f is a new rule, separate from the first exercise's square function. To undo multiplying by three and subtracting two, add two and then divide by three. Solving y = 3x − 2 for x gives x = (y + 2)/3. The inverse is f⁻¹(x) = (x + 2)/3.
For example, input 4 gives output 10, and applying the inverse to 10 returns 4. For every real input, ((3x − 2) + 2)/3 = x and 3((y + 2)/3) − 2 = y check both directions. The original domain and the range of outputs it actually reaches are both all real numbers. The inverse therefore also accepts every real number.
The symbol f⁻¹ names the function that returns from an output to the original input. It differs from the reciprocal 1/f(x). For the preceding f, f⁻¹(2) = 4/3 whereas 1/f(2) = 1/4. Moreover, 1/(3x − 2) is undefined at x = 2/3, while the inverse (x + 2)/3 accepts that input.
If a candidate inverse looks plausible, feed it the original function's output and check that it returns the original input. The exercise's candidate (x − 2)/3 instead gives x − 4/3 after f. The candidate 3x + 2 gives 9x − 4. Check that division undoes multiplication by three as well as checking the sign of the constant.
The third exercise asks for an even function on all real numbers: one satisfying f(−x) = f(x) for every real x. This does not restrict inputs to even integers. The choice x⁴ + 1 satisfies (−x)⁴ + 1 = x⁴ + 1. The other choices, 2x, x³ and x + 1, give different values at 1 and −1, so none is even.
The rule x⁴ + 1 sends both 1 and −1 to 2. From output 2, a reverse rule cannot choose one input and recover both originals. Thus this function on the whole real domain has no inverse that returns every original input. Squaring has the same problem. Restricting the square function to nonnegative inputs instead gives nonnegative outputs and allows the nonnegative square root to recover the original input. Retain that change of domain when describing the inverse.
For paper practice, set A(x) = 2x + 3 and B(x) = x². Draw separate arrows for A(B(2)) and B(A(2)), then find the inverse of A. The first chain is 2 → 4 → 11; the second is 2 → 7 → 49. The inverse of A is (x − 3)/2, which returns its output to the original input.
The three exercises below cover composition, inversion and evenness in that order. Write the intermediate value, the reverse calculation and the sign-reversed input before opening the answers. For substitution, revisit the prerequisite “Expressions: give the unknown a role.” For symmetry arising from a square, read the related “Completing the square: minima, symmetry and real solutions.” The practice preset selects standard, high-school-level questions on expressions and functions, including other matching questions beyond these three.
TRY & READ
Compare the choices before opening the answer and explanation. Reading an example does not save a test answer or score.
Example 1 · Algebra and functions
Answer: 9
Apply the inner function first: g(2) = 3. Then f(3) = 3² = 9.
Example 2 · Algebra and functions
Answer: f⁻¹(x)＝(x＋2)/3
Solve y = 3x − 2 for x: x = (y + 2)/3. Swapping input and output labels gives f⁻¹(x) = (x + 2)/3. An inverse function is different from a reciprocal.
Example 3 · Algebra and functions
Answer: f(x)＝x⁴＋1
An even function satisfies f(−x) = f(x). For x⁴ + 1, the fourth power is unchanged by reversing the sign of x. The other choices fail this identity.
Source checked: 2026-10-21
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