Functions and function notation

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Apuntes de la lección Diapositivas Preguntas de práctica

Apuntes de la lección

What a function is

A function is a rule that takes an input and gives exactly one output. A function machine shows the steps.
Function notation: f(x) = 3x + 2 means "the function f acts on x". Read f(x) as "f of x", not f times x.
To evaluate f(4), replace x with 4: f(4) = 3(4) + 2 = 14.

Composite functions

fg(x) means "do g first, then f": fg(x) = f(g(x)).
If f(x) = 2x and g(x) = x + 3, then fg(x) = 2(x + 3) = 2x + 6, but gf(x) = 2x + 3. The order matters.

Composition: g first, then f

Inverse functions

The inverse function f^-1(x) reverses f. To find it: write y = f(x), swap x and y, then rearrange for y.
If f(x) = 2x + 5, then x = 2y + 5, so y = \frac{x - 5}{2} and f^-1(x) = \frac{x - 5}{2}.
f^-1(f(x)) = x: applying a function and its inverse returns the starting value.
An inverse must return exactly one input for each output. For example, x² needs a domain restriction such as x ≥ 0 to have inverse √x.

Domain and range

The domain is the set of allowed inputs; the range is the set of possible outputs.
f(x) = \frac{1}{x} cannot take x = 0, and f(x) = \√{x} needs x \geq 0.

Solving using functions

To solve f(x) = 11 when f(x) = 3x + 2, write 3x + 2 = 11, so x = 3.

Diapositivas

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Preguntas de práctica

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1.In function notation, what does f(x) mean?
Easy
- Af multiplied by x
- BThe output of function f for input x
- CThe factors of x
- Df divided by x
2.If f(x) = 2x + 3, what is f(4)?
Easy
- A8
- B11
- C14
- D24
3.If f(x) = x² - 1, what is f(5)?
Easy
- A9
- B24
- C26
- D4
4.If f(x) = 10 - x, what is f(-2)?
Easy
- A8
- B12
- C-8
- D-12
5.If f(x) = 4x, what is f(x + 1)?
Easy
- A4x + 1
- B4x + 4
- C4x
- Dx + 4
6.If f(x) = 3x + 2 and f(a) = 14, what is a?
Easy
- A4
- B5.33
- C12
- D3
7.If f(x) = 2x and g(x) = x + 3, what is fg(x)?
Easy
- A2x + 3
- B2x + 6
- C2x² + 3
- Dx + 6
8.If f(x) = 2x and g(x) = x + 3, what is gf(x)?
Easy
- A2x + 3
- B2x + 6
- C2x² + 3
- Dx + 6

Más preguntas

9.If f(x) = x + 4 and g(x) = 3x, what is fg(2)?
Medium
- A10
- B18
- C6
- D14
10.If f(x) = x² and g(x) = x - 3, what is gf(4)?
Medium
- A1
- B13
- C25
- D16
11.Find f^-1(x) when f(x) = x + 7.
Medium
- Ax + 7
- Bx - 7
- C7 - x
- D\frac{x}{7}
12.Find f^-1(x) when f(x) = 2x + 5.
Medium
- A\frac{x - 5}{2}
- B\frac{x}{2} + 5
- C2x - 5
- D\frac{x + 5}{2}
13.Find f^-1(x) when f(x) = \frac{x}{3} - 2.
Medium
- A3x + 2
- B3(x + 2)
- C3x - 2
- D\frac{x + 2}{3}
14.If f(x) = 3x - 1, what is f^-1(8)?
Medium
- A23
- B3
- C2.33
- D1
15.For f(x) = 2/x with nonzero real inputs, find the inverse function.
Medium
- A\frac{x}{2}
- B\frac{2}{x}
- C-\frac{2}{x}
- D2x
16.Which value of x cannot be used in f(x) = \frac{1}{x - 3}?
Easy
- A0
- B1
- C3
- D-3
17.What is the domain restriction of f(x) = \√{x} for real outputs?
Easy
- Ax > 0 only
- Bx \geq 0
- Cx \leq 0
- Dnone
18.If f(x) = x² + 1, what is the range for real x?
Medium
- Af(x) \geq 0
- Bf(x) \geq 1
- Cf(x) \leq 1
- Dall numbers
19.If f has an inverse function and x is in its domain, what is f⁻¹(f(x)) equal to?
Easy
- A0
- B1
- Cx
- Df(x)
20.If f(x) = 2x + 1 and g(x) = x², solve fg(x) = 19.
Medium
- Ax = \pm 3
- Bx = 3 only
- Cx = \pm 9
- Dx = 9
21.A function machine multiplies by 3, then subtracts 4. Which function is it?
Medium
- Af(x) = 3(x - 4)
- Bf(x) = 3x - 4
- Cf(x) = 3 - 4x
- Df(x) = 3x + 4
22.What is the inverse of a function machine step 'multiply by 5'?
Easy
- AMultiply by 5
- BAdd 5
- CDivide by 5
- DSubtract 5
23.If f(x) = x², what is f(-3)?
Medium
- A-9
- B9
- C6
- D-6
24.If f(x) = 5 - 2x, what is f^-1(x)?
Medium
- A\frac{5 - x}{2}
- B\frac{x - 5}{2}
- C5 - \frac{x}{2}
- D\frac{x + 5}{2}
25.Which of these is a function?
Easy
- AOne input giving two outputs
- BOne input giving exactly one output
- CTwo inputs giving no output
- DAn equation with no variable
26.f(x) means f multiplied by x.
Easy
True or false?
27.fg(x) means apply g first and then f.
Easy
True or false?
28.fg(x) always equals gf(x).
Easy
True or false?
29.The inverse of f(x) = x - 5 is f^-1(x) = x + 5.
Easy
True or false?
30.f(x) = \frac{1}{x} is defined at x = 0.
Medium
True or false?
31.Which of these are correct for f(x) = 2x + 1?
Medium
- Af(3) = 7
- Bf(0) = 0
- Cf(-1) = -1
- Df^-1(x) = \frac{x - 1}{2}
- Ef(2) = 6
32.Which value is excluded from the domain of f(x) = \frac{1}{x - 2}?
Medium
- Ax = 2
- Bx = 0
- Cx = 1
- Dx = -2
33.Find the inverse of f(x) = 4x - 3, showing the method.
Medium

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