1.1.4 Practice set: functions

Domain and range

Domain and range (lesson 1.1.1)

0 of 6 correct

1.
If $f(x) = 3 - x^2$ , find $f(4)$
2.
Which input is not in the domain of $\dfrac{1}{x + 5}$ ?
3.
What is the largest input in the domain of $\sqrt{12 - 3x}$ ?
4.
What is the smallest number in the range of $(x - 1)^2 + 3$ ?
5.
Which input gives $\sqrt{2x + 1}$ the output 5?
6.
The domain of $\dfrac{1}{\sqrt{x + 2}}$ is $x > c$ . What is $c$ ?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Transformations and pieces

Transformations and pieces (lesson 1.1.2)

0 of 7 correct

1.
The graph of $y = x^2$ is moved 1 left and 2 down. Write the new formula
2.
$\sqrt{x}$ is moved 2 to the left. What is the smallest input in the new domain?
3.
$f(x) = x^2 + x$ . Write $-f(-x)$ as a formula
4.
$x^2$ reaches 9 at $x = 3$ . At which positive $x$ does $(3x)^2$ reach 9?
5.
$g(x) = 5$ when $x \le 2$ and $g(x) = 3x - 1$ when $x > 2$ . Find $g(2)$
6.
Evaluate $\max(12 - 10, 0) + \max(10 - 12, 0)$
7.
Evaluate $-x^2$ at $x = 3$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Composition and inverses

Composition and inverses (lesson 1.1.3)

0 of 7 correct

1.
$f(x) = x + 4$ and $g(x) = 3x$ . Find $f(g(2))$
2.
With the same $f$ and $g$ , find $g(f(2))$
3.
$f(x) = \sqrt{x}$ and $g(x) = x + 9$ . Find $f(g(16))$
4.
$f(x) = \sqrt{x}$ and $g(x) = 6 - x$ . What is the largest input of $f(g(x))$ ?
5.
Find the inverse of $f(x) = 2(x - 3)$
6.
$f(x) = 4x - 2$ . Find $f^{-1}(10)$
7.
$f(x) = (x - 1)^2$ is kept to $x \ge 1$ . Find $f^{-1}(16)$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Functions that build functions

Write two functions. Both return a function, as compose did in lesson 1.1.3.

compose(outer, inner) returns the function $x \mapsto \text{outer}(\text{inner}(x))$ .
inverse_linear(a, b) returns the inverse of $f(x) = ax + b$ . Raise ValueError when a == 0, because a flat line has no inverse.

The tests call your results with numbers and with NumPy arrays.

See the tests

import numpy as np


def _double_plus_one(x):
    return 2 * x + 1


def _square(x):
    return x**2


def test_compose_returns_function():
    """compose(f, g) returns something you can call"""
    h = compose(_double_plus_one, _square)
    assert callable(h), f"expected a function, got {h!r}"


def test_compose_inner_first():
    """compose(f, g)(3) applies g first: f(g(3)) = 19"""
    expect_close(compose(_double_plus_one, _square)(3), 19, what="compose(f, g)(3)")


def test_compose_order_matters():
    """compose(g, f)(3) is g(f(3)) = 49"""
    expect_close(compose(_square, _double_plus_one)(3), 49, what="compose(g, f)(3)")


def test_compose_arrays():
    """Composition works on a NumPy array"""
    x = np.array([0.0, 1.0, 2.0])
    expect_close(compose(_double_plus_one, _square)(x), 2 * x**2 + 1, what="compose(f, g) on [0, 1, 2]")


def test_inverse_simple():
    """The inverse of 2x + 3 sends 13 back to 5"""
    expect_close(inverse_linear(2, 3)(13), 5.0, what="inverse_linear(2, 3)(13)")


def test_inverse_round_trip():
    """Going there and back returns the input, for f(x) = −0.5x + 4"""
    x = np.array([-3.0, 0.0, 1.5, 8.0])
    expect_close(inverse_linear(-0.5, 4)(-0.5 * x + 4), x, what="f⁻¹(f(x)) for several x")


def test_flat_line_has_no_inverse():
    """inverse_linear(0, 7) raises ValueError"""
    try:
        inverse_linear(0, 7)
    except ValueError:
        return
    raise AssertionError("expected ValueError for a == 0")

Domain and range

Transformations and pieces

Composition and inverses

Coding exercise

Free further reading