Generalities on functions

What Nour says

1. 1. Reminders

   Today, the whole chapter: "Generalities on functions", Year 3, Mathematics section.

   We will cover six parts: reminders, restriction of a function, bounds, piecewise affine functions, the square root function, and operations on functions.

   Let's start with reminders about functions, the ideas we already know.

   A function takes a number x and gives it exactly one value, called f of x.

   The set of all x where f of x can be computed is called the domain of definition (ensemble de définition).

   That is, the domain of definition, the set of all allowed values of x for the function.

   We often draw this function with a curve, in a coordinate plane.

   Let's recall how to find a domain, with a worked example.

   Here f of x is defined when the denominator is not zero.

   So x must not equal two.

   The domain is all real numbers except two.

   Let's plot this function to see its shape near that forbidden point.

   Near x equals two, the curve shoots up or down, since division there is impossible.

   Now let's recall another idea from before: parity, which describes a kind of symmetry.

   That is, parity tells us if the curve is symmetric about the y-axis or about the origin.

   A function is even when f of minus x equals f of x, for every x in the domain.

   A function is odd when f of minus x equals minus f of x.

   An even function has a curve symmetric about the vertical axis.

   An odd function has a curve symmetric about the origin point.

   Before checking parity, we must always check one thing first about the domain.

   The domain itself must be symmetric around zero, or parity makes no sense.

   Now let's work through a full example, checking parity of a given function.

   Consider the function f of x equals x squared minus three.

   First, the domain of f is all real numbers, since this formula always works.

   This domain is symmetric, since for any x, minus x is also real.

   Now we compute f of minus x, replacing x by minus x everywhere.

   Minus x squared is the same as x squared, so this simplifies.

   Since f of minus x equals f of x, this function is even.

   Let's draw this parabola, to see the symmetry with our own eyes.

   Here is the curve of f, shaped like a U, centered at the y-axis.

   The left side mirrors the right side perfectly, across the vertical axis.

   At x equals two and x equals minus two, f gives the same value, one.

   So the big ideas here are the domain, and parity, even or odd.

   Remember: always check the domain is symmetric before testing parity.

   And remember: even means same value, odd means opposite value, at minus x.

   Is the function g of x equal to x cubed even, odd, or neither?

2. 2. Restriction of a function

   Now let's talk about the restriction of a function to an interval.

   That is, we take the same function but only look at it on a smaller interval.

   We write f restricted to I, meaning f but with its domain cut down to I.

   The rule stays exactly the same, only the allowed inputs change.

   The interval I must be inside the original domain of f.

   Here is why this matters: a function can behave very differently on different intervals.

   Take f of x equals x squared, defined on all real numbers.

   On the whole line, this function goes down then up, not always the same direction.

   But if we restrict it to the interval zero, plus infinity, it only goes up.

   That restricted piece is increasing, which is a cleaner, simpler behavior to study.

   Let's see a full worked example, finding a restriction and its new domain.

   Consider f of x equals one over x, with domain all reals except zero.

   We want the restriction of f to the interval one, plus infinity.

   First we check that I sits fully inside the domain of f.

   Since zero is not in I, this restriction is perfectly valid.

   The restriction is written f restricted to I, with the same formula, one over x.

   Let's draw this restricted piece, only the part of the curve from x equals one onward.

   Here is the full curve of one over x, in gray, for reference.

   And here, in green, is only the restricted part, starting at x equals one.

   Notice this restricted piece only decreases, unlike the full curve with two branches.

   So restriction means: same formula, smaller domain, often simpler behavior.

   Always check first that your chosen interval lies inside the original domain.

   Then the restricted function keeps the same rule, just on fewer inputs.

   What is f restricted to the interval minus infinity, zero, for f of x equal one over x?

3. 3. Upper bound - Lower bound

   Now let's study bounds of a function: majorant and minorant.

   A majorant is a number that the function never goes above, on its domain.

   That is, f of x stays less than or equal to M, for every x in I.

   A minorant works the other way: the function never goes below it.

   That is, f of x stays greater than or equal to m, for every x in I.

   When a function has both, we call it bounded on that interval.

   Here is why this matters: bounds tell us the function stays in a band, never escaping.

   Picture a horizontal strip around the curve, a ceiling line and a floor line.

   The curve must stay below the ceiling, and above the floor, everywhere on I.

   Note that a majorant is never unique, any bigger number also works as one.

   Let's see a full worked example, finding bounds of a given function.

   Consider f of x equals x squared minus two x plus three, on all real numbers.

   Let's rewrite this using the canonical form, completing the square.

   A square is always greater than or equal to zero, for any real number.

   So adding two to both sides, f of x is at least two.

   So two is a minorant of f on all real numbers.

   There is no majorant here, since the square grows without any limit.

   Let's draw this parabola to see its lowest point clearly.

   Here is the curve, a U shape opening upward, with its bottom at two.

   The lowest point is at x equals one, where f of x equals two exactly.

   The floor line at height two touches the curve but the curve never dips below.

   So two is the best possible minorant, the biggest number that still works.

   No number works as a ceiling, since the curve rises forever.

   So the big idea: a minorant bounds f from below, a majorant bounds it from above.

   Finding these bounds often means rewriting the formula, like completing the square.

   Remember: a function can have a minorant without having any majorant.

   Is the function h of x equal to minus x squared bounded above or below on all reals?

4. 4. Piecewise affine functions

   Now let's study functions that are affine on each separate piece of their domain.

   That is, we cut the domain into pieces, and on each piece f looks like a line.

   On another piece, f can use a different line, with different slope and intercept.

   Each piece is a straight segment or ray, but they can meet or break.

   Here is why this matters: many real situations change rule depending on the range.

   Think of a taxi price, flat at first, then rising steadily with distance.

   The graph of such a function looks like connected straight segments, like a broken line.

   The absolute value function is the simplest example: a line that bends once, at zero.

   Let's see a full worked example, studying a function given by two pieces.

   Consider f of x equal to x plus one when x is at least minus one.

   And f of x equal to minus x minus one when x is less than minus one.

   First let's check both pieces agree at the breaking point, x equals minus one.

   And the other piece: minus minus one minus one also gives zero.

   Both pieces meet exactly at the point minus one, zero, so the curve is connected.

   Let's draw this, piece by piece, on our coordinate plane.

   For x at least minus one, we draw the line x plus one, rising steadily.

   For x less than minus one, we draw the line minus x minus one, falling as x grows.

   Both pieces meet at the point minus one, zero, forming a sharp corner there.

   This whole shape is actually the absolute value of x plus one.

   So this function is affine on each piece, but overall it forms a V shape.

   The method: identify each piece's interval, write its own line formula, then check the joins.

   Checking the joins tells us whether the graph connects smoothly or breaks apart.

   Remember: at the breaking point, plug the value into both formulas and compare.

   If both formulas agree there, the graph has no gap, just a corner or a smooth join.

   What shape does the graph of f of x equal absolute value of x minus two make?

5. 5. The function x ↦ √g(x)

   Now let's study functions built from a square root, like x mapped to the square root of g of x.

   A square root only exists for numbers that are zero or positive, never negative.

   So the domain of this new function is all x where g of x is at least zero.

   We must also stay inside the domain of g itself, before even checking the sign.

   Here is why this matters: the square root symbol silently adds a condition on x.

   Picture g of x as a line crossing zero somewhere, splitting the axis into two parts.

   Only where this line sits above the x-axis can we take its square root.

   Let's see a full worked example, finding the domain of such a function.

   Consider f of x equal to the square root of three minus x.

   Here g of x is three minus x, and it must be zero or positive.

   Adding x to both sides, we get three is at least x.

   In other words, x must be less than or equal to three.

   So the domain is the interval from minus infinity up to three, included.

   Let's draw this function, to see where the curve actually exists.

   Here is the curve, only appearing for x values up to three.

   At x equals three, the curve touches zero, then nothing exists beyond that point.

   So the method: set the inside of the square root greater than or equal to zero.

   Then solve that inequality, and intersect it with the domain of g.

   Remember: the square root always forces this extra condition on the domain.

   What is the domain of f of x equal to square root of x minus five?

6. 6. Operations on functions

   Now let's study operations on functions: adding, multiplying, and dividing them.

   That is, we can build new functions by combining two existing functions with basic operations.

   We can also multiply two functions together, point by point.

   Or divide one function by another, when the bottom one is not zero.

   Each new function has its own domain, built from the domains of f and g.

   Here is why this matters: the new domain is where both original functions already work.

   For division, we also remove any x where g of x becomes zero.

   Let's see a full worked example, combining two concrete functions.

   Let f of x equal square root of x, and g of x equal x minus one.

   The domain of f is all x at least zero, since square roots need nonnegative numbers.

   The domain of g is all real numbers, since it is a simple line.

   We want the domain of f divided by g, so we combine both conditions.

   First, intersecting the two domains gives us x at least zero.

   Next, g of x equals zero exactly when x equals one.

   So we must remove the point one from our interval.

   Let's draw this on a number line, to see the removed point clearly.

   Here is the ray starting at zero, going right, with a hole at one.

   The open circle at one shows that single point is excluded from the domain.

   Everywhere else on this ray, both f divided by g and its formula work fine.

   So the method: find each domain separately, intersect them, then remove zeros of the denominator.

   This same idea applies whether we add, multiply, or divide two functions together.

   Remember: for division only, we add the extra condition that the denominator is not zero.

   That finishes our six parts of this chapter on generalities about functions.

   What is the domain of f plus g, if f of x equals square root of x and g of x equals one over x?

   That completes our full chapter on generalities about functions.

   We covered domains, parity, restrictions, bounds, piecewise affine functions, square roots, and operations.

   Practice the two exercises to make sure these ideas feel solid.