Functions and function notation
边玩边学
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课程笔记
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.
幻灯片
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练习题
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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)?
Easy3.If f(x) = x2 - 1, what is f(5)?
Easy4.If f(x) = 10 - x, what is f(-2)?
Easy5.If f(x) = 4x, what is f(x + 1)?
Easy6.If f(x) = 3x + 2 and f(a) = 14, what is a?
Easy7.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
更多题目
9.If f(x) = x + 4 and g(x) = 3x, what is fg(2)?
Medium10.If f(x) = x2 and g(x) = x - 3, what is gf(4)?
Medium11.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)?
Medium15.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}?
Easy17.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?
Easy20.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)?
Medium24.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.
EasyTrue or false?
27.fg(x) means apply g first and then f.
EasyTrue or false?
28.fg(x) always equals gf(x).
EasyTrue or false?
29.The inverse of f(x) = x - 5 is f-1(x) = x + 5.
EasyTrue or false?
30.f(x) = \frac{1}{x} is defined at x = 0.
MediumTrue 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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