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Functions · Real functions

Characteristics of functions

Analysing a function completely means always answering the same questions: where is it defined?, where does it cross the axes?, where is it positive or negative?, what values can it take? and does it have any symmetry? In Bachillerato these questions are combined in more complex functions than in ESO.

Before you start, make sure you know: Inequalities in one unknown (4th ESO)
Before you start, make sure you know: Basic concepts (4th ESO)

Domain with several restrictions at once

Strip diagram showing the intersection of the region where the radicand is non-negative and the region where the denominator is non-zero. radicand ≥ 0 denominator ≠ 0 final domain: the intersection
Practice★☆☆
Determine the domain of the rational function $f(x) = \frac{x-1}{x^2-9}$.

Determine the domain of the rational function $f(x) = \frac{x-1}{x^2-9}$.

Solve
Practice★★☆
Find the domain of the function $f(x) = \sqrt{\frac{x+1}{x-3}}$.

Find the domain of the function $f(x) = \sqrt{\frac{x+1}{x-3}}$.

Solve
Practice★★☆
Determine the domain of the function $f(x) = \ln(x^2 - 4x + 3)$.

Determine the domain of the function $f(x) = \ln(x^2 - 4x + 3)$.

Solve

Intercepts and sign analysis

Practice★☆☆
Determine the intervals in which the function $q(x) = -3x + 12$ is positive and those in which it is negative.

Determine the intervals in which the function $q(x) = -3x + 12$ is positive and those in which it is negative.

Solve
Practice★★☆
Find the axis intercepts and study the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.

Find the axis intercepts and study the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.

Solve

Range of a function

Graph of an upward-opening parabola with vertex at (0, 4), highlighting on the vertical axis the half-line [4, +∞) that corresponds to its range. y = 4 (minimum of the range)
Practice★☆☆
State the range of the quadratic function $r(x) = x^2 + 4$.

State the range of the quadratic function $r(x) = x^2 + 4$.

Solve
Practice★★★
Find the inverse function of $f(x) = \frac{2x-5}{x+1}$ and use it to determine the range of the function $f(x)$.

Find the inverse function of $f(x) = \frac{2x-5}{x+1}$ and use it to determine the range of the function $f(x)$.

Solve

Symmetry: even and odd functions

Graph symmetric about the vertical axis showing the equality of heights f(-x) = f(x) for two symmetric values on the horizontal axis. x−x
Practice★★☆
Determine analytically whether the function $h(x) = \frac{x^3}{x^2+1}$ is even, odd or has no symmetry.

Determine analytically whether the function $h(x) = \frac{x^3}{x^2+1}$ is even, odd or has no symmetry.

Solve
Practice★★★
Prove analytically that the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$ is odd and determine its domain.

Prove analytically that the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$ is odd and determine its domain.

Solve

Summary

Let's practice! 10 medium exercises in this section
★★☆ Find the domain of the function $f(x) = \sqrt{\frac{x+1}{x-3}}$.★★☆ Find the axis intercepts and study the sign of the rational function…★★☆ Determine analytically whether the function $h(x) = \frac{x^3}{x^2+1}$ is even, odd or…★★☆ A projectile is launched vertically upwards. The height $h(t)$ in metres that it reaches…
Let's practice! 10 hard exercises in this section
★★★ Find the domain of the function $f(x) = \sqrt{\log_2(x-3)}$, justifying your answer.★★★ Determine the coefficients $a$, $b$ and $c$ of the quadratic function…★★★ The average monthly cost (in euros) per unit produced in an electronic components factory…
See all the exercises in this section