Lesson 2 — Functions: definition, domain, image

Determine the maximum domain of $f(x) = 3x + 1$.

Determine the maximum domain of $g(x) = \dfrac{1}{x - 2}$.

Determine the maximum domain of $h(x) = \sqrt{x - 5}$.

Determine the maximum domain of $f(x) = \dfrac{1}{(x+2)(x-3)}$.

Let $f(x) = 2x^2 + 2$. Compute $f(2)$, $f(-1)$, $f(0)$.

Is the function $f(x) = 3x - 1$ injective? Justify.

Is the function $g(x) = x^2$ defined on $\mathbb{R}$ injective?

What is the image of $g(x) = x^2$ defined on $\mathbb{R}$?

For the piecewise function $f(x) = \begin{cases} 3x + 1 & \text{if } x < 0 \\ x^2 + 3 & \text{if } x \geq 0 \end{cases}$ compute $f(-3)$ and $f(2)$.

Determine the domain and image of $f(x) = \sqrt{4 - x^2}$.

Let $f(x) = x^2$ and $g(x) = x + 1$. Compute $(f \circ g)(x)$.

With the same $f, g$ as above, compute $(g \circ f)(x)$.

Determine the inverse of $f(x) = 3x + 1$.

Why does $f(x) = x^2$ defined on $\mathbb{R}$ have no inverse? What about on $[0, +\infty)$?

Is the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3$ bijective?

Is the function $g: \mathbb{R} \to [0, +\infty)$ defined by $g(x) = x^2$ surjective? And injective?

Let $f(x) = x^2$ and $g(x) = 2x + 3$. Compute $(f \circ g)(x)$ and $(g \circ f)(x)$ and show they are different.

Determine $f(x)$ given that $(f \circ g)(x) = 3x + 4$ and $g(x) = x - 1$.

Let $f, g: \mathbb{R} \to \mathbb{R}$ be functions such that $(f \circ g)(x) = x^2 + 1$ and $g(x) = x + 1$. Determine $f(x)$.

Prove: the composition of two bijective functions is bijective.

Determine the maximum domain of $f(x) = \dfrac{x+1}{x^2 - 9}$.

Determine the maximum domain of $f(x) = \sqrt{4 - x}$.

Determine the domain of $f(x) = \sqrt{4 - x^2}$.

Determine the domain of $f(x) = \dfrac{1}{\sqrt{x - 2}}$.

Determine the domain of $f(x) = \dfrac{\sqrt{9 - x^2}}{x}$.

Use the horizontal line test to decide whether $f(x) = x^3 - 3x$ is injective on $\mathbb{R}$.

Is the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3$ injective? Surjective? Bijective?

Let $f(x) = 2x + 3$ and $g(x) = x^2$. Compute $f \circ g$, $g \circ f$, $f \circ f$, $g \circ g$.

Determine $f(x)$ given that $(f \circ g)(x) = 4x^2 - 4x + 5$ and $g(x) = 2x - 1$. (Hint: let $u = 2x - 1$.)

Sketch $f(x) = |x - 3|$ starting from the graph of $|x|$. What transformation occurred?

Sketch $f(x) = -2(x+1)^2 + 4$ using transformations of $x^2$.

Decide whether each function below is even, odd, or neither: (a) $f(x) = x^4 - x^2$; (b) $g(x) = x^3 + x$; (c) $h(x) = x^2 + x$.

Consider the characteristic function $\chi_A(x) = 1$ if $x \in A$, $0$ otherwise. For $A = [0, 1]$, determine the domain and image.

Verify that $\sin x$ has period $2\pi$. Is there a smaller period?

Compute the Euclidean distance between $(1, 2)$ and $(5, 7)$.

To shift the graph of $y = f(x)$ two units downward, which transformation applies?

To shift $y = f(x)$ three units to the right, write $g(x) =$ ?

Determine the domain, image, and classify $f(x) = x^3 + 3x + 1$.

Differentiate the vertical stretch $g(x) = 2 f(x)$ from the horizontal stretch $g(x) = f(2x)$.

Determine the domain and image of $f(x) = 1/x$.

A taxi charges R$ 5.50 fixed plus R$ 3.10 per km. (a) Write the cost function $T(d)$. (b) How much does a 12 km trip cost? (c) For what distance is the cost R$ 80?

An empty pool is filled at 200 L/min. Model $V(t)$ in liters as a function of time $t$ in minutes. Total capacity 8000 L. Determine the physical domain and image.

Compute the BMI of a person weighing 70 kg and 1.75 m tall. In which WHO range do they fall?

A factory produces $q$ units per day with cost $C(q) = 100 + 8q + 0.1q^2$ reais. (a) Fixed cost? (b) Average cost at $q = 50$? (c) Marginal cost of the 51st unit?

A bacterium doubles every 30 min. Model $N(t)$ if $N(0) = 100$.

The recommended maximum heart rate is $F_{max}(\text{age}) = 220 - \text{age}$. Compute it for ages 30, 50, 70.

The function $V(t) = 30\,000 \cdot (0.85)^t$ models the resale value of a car $t$ years after purchase. (a) $V(0)$? (b) $V(5)$? (c) For what $t$ does the value fall below R$ 10,000?

Model mathematically: "the sum of two numbers is 30 and the product is maximum". (Preview of quadratics — Lesson 4.)

In a factory, each worker assembles 12 products/day. Beyond 50 workers, each additional worker assembles only 8 products. Model $P(n)$ as a piecewise function.

A rectangular pool has a fixed perimeter of 30 m. Model the area $A$ as a function of the length $\ell$. Determine the physical domain and the maximum area.