Lesson test: Functions and inverses

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Functions and inverses

1.[2p]
What exactly does the definition of a function demand?

Each input in the domain has exactly one output

Correct

Each output comes from exactly one input

The answer is: Each input in the domain has exactly one output

The rule can be written as a formula

The answer is: Each input in the domain has exactly one output

The domain and the range are the same set

The answer is: Each input in the domain has exactly one output

2.[2p]
The domain of $f (x) = 1 / (x - 2)$ is every real number except one. Which?

CorrectNot quite: 2

3.[2p]
With $f (x) = 2 x - 5$ and $g (x) = x^{2} + 1$ , what is $g (f (3))$ ?

CorrectNot quite: 2

4.[2p]
With the same $f$ and $g$ , what is $f (g (3))$ ?

CorrectNot quite: 15

5.[3p]
Why does $f (x) = x^{2}$ on all of the reals have no inverse?

It is not injective: $3$ and $- 3$ have the same output, so the inverse would need two values at $9$

Correct

Its range is not all of the reals

The answer is: It is not injective: $3$ and $-3$ have the same output, so the inverse would need two values at $9$

Because squaring is not associative

The answer is: It is not injective: $3$ and $-3$ have the same output, so the inverse would need two values at $9$

Because its domain excludes the negatives

The answer is: It is not injective: $3$ and $-3$ have the same output, so the inverse would need two values at $9$

6.[1p]
$f^{- 1} (x)$ means $1 / f (x)$ .

True

The answer is: False

False

Correct

7.[3p]
United Kingdom income tax in 2024/25 charges nothing below 12,570 pounds and 20 per cent from there to 50,270. What is the tax on an income of 30,000 pounds?

CorrectNot quite: 3486

8.[2p]
Put the steps for finding an inverse algebraically in order.

Gather every term containing $x$ on one side
Rename the variables so the inverse is written in $x$
Solve for $x$ in terms of $y$
Write $y = f (x)$

Show the answer

a, b, c, d

9.[3p]
Match each function to its inverse.

$3 x + 1$
$x^{3}$
$9 c / 5 + 32$
$x^{2}$ with $x \geq 0$

the positive square root of $x$
$5 (x - 32) / 9$
the cube root of $x$
$(x - 1) / 3$

Show the answer

$3 x + 1$ : $(x - 1) / 3$ $x^{3}$ : the cube root of $x$ $9 c / 5 + 32$ : $5 (x - 32) / 9$ $x^{2}$ with $x \geq 0$ : the positive square root of $x$

1.[2p] What exactly does the definition of a function demand?

2.[2p] The domain of f(x)=1/(x-2) is every real number except one. Which?

3.[2p] With f(x)=2x-5 and g(x)=x2+1, what is g(f(3))?

4.[2p] With the same f and g, what is f(g(3))?

5.[3p] Why does f(x)=x2 on all of the reals have no inverse?

6.[1p] f-1(x) means 1/f(x).

7.[3p] United Kingdom income tax in 2024/25 charges nothing below 12,570 pounds and 20 per cent from there to 50,270. What is the tax on an income of 30,000 pounds?

8.[2p] Put the steps for finding an inverse algebraically in order.

9.[3p] Match each function to its inverse.

1.[2p]
What exactly does the definition of a function demand?

2.[2p]
The domain of $f (x) = 1 / (x - 2)$ is every real number except one. Which?

3.[2p]
With $f (x) = 2 x - 5$ and $g (x) = x^{2} + 1$ , what is $g (f (3))$ ?

4.[2p]
With the same $f$ and $g$ , what is $f (g (3))$ ?

5.[3p]
Why does $f (x) = x^{2}$ on all of the reals have no inverse?

6.[1p]
$f^{- 1} (x)$ means $1 / f (x)$ .

7.[3p]
United Kingdom income tax in 2024/25 charges nothing below 12,570 pounds and 20 per cent from there to 50,270. What is the tax on an income of 30,000 pounds?

8.[2p]
Put the steps for finding an inverse algebraically in order.

9.[3p]
Match each function to its inverse.