令f:A-> B和g:B-> C是两个函数。然后,用gof表示的f和g的组成定义为函数gof:由gof(x)= g {f(x)}给出的A-> C,x∈A。
显然,dom(gof)= dom(f)。
此外,仅当range(f)是dom(g)的子集时才定义gof。
评估复合函数
我们知道复合函数写为fog(x),gof(x)等。在这里,fog(x)将被评估为f {g(x)},gof(x)将被评估为g {f(x)}。
问题1:让f:{2,3,4,5}-> {3,4,5,9}和g:{3,4,5,9}-> {7,11,15}被定义为函数作为
f(2)= 3,f(3)= 4,f(4)= f(5)= 5且g(3)= g(4)= 7和g(5)= g(9)= 11。找出gof(x)。
解决方案:
g o f(x) = g{ f(x)}. So first we find the inner bracket which is f(x) here.
We have the values of f(2), f(3), f(4) and f(5), hence we have to find the values of g o f(x) for all these values.
g o f(2) = g{ f(2) } = g(3) = 7,
g o f (3) = g{ f(3) } = g(4) = 7,
g o f(4) = g{ f(4) } = g(5) = 11
And g o f(5) = g(5) = 11.
问题2:表明如果f:A-> B和g:B-> C在上面,那么gof:A-> C也在上面。
解决方案
Given an arbitrary element z ∈ C, there exists a pre-image y of z under g such that g(y) = z, since g is onto.
Further, for y ∈ B, there exists an element x in A with f(x) = y, since f is onto.
Therefore, g o f(x) = g{ f(x) } = g(y) = z, showing that g o f is onto.
问题3:让ƒ:R-> R:f(x)=(x 2 – 3x + 2)。找出fof(x)。
解决方案:
f o f(x) = f {f(x) } = f(x2 – 3x + 2) = f(y) (let y = x2 – 3x + 2)
= y2 – 3y + 2
= (x2 – 3x + 2) – 3(x2 – 3x + 2) + 2
= x4 – 6x3 + 10x2 – 3x.
评估复合函数:使用表
在这种类型的问题中,我们将得到一个具有x,f(x),g(x)值的表,并且需要像示例1中那样找到f(x)和g(x)的组合。要求找到雾气(1)。
为了找到解决方案,我们将从内括号开始,并在给定的表中找到它的值,因此我们将从表中找到g(1)的值为4。然后对于外括号,我们将再次找到它的值从给定的表中,所以我们将从表中找到f(4)的值为2。因此,在这种情况下,我们将获得所需的答案,即2。
问题1:使用下表,评估fog(1)和gof(4)。
x | f(x) | g(x) |
---|---|---|
1 | 5 | 4 |
2 | 8 | 5 |
3 | 4 | 3 |
4 | 2 | 8 |
解决方案:
So, f o g(1) = f{g(1)} and g o f(4) = g{f(4)}.
We first evaluate the inner bracket then the outer bracket, using the values given in table.
For f{g(1)}, g(1) = 4. Now f(4) = 2.
Hence, f{g(1)} = 2.
Similarly, for g{f(4)}, f(4) = 2. Now g(2) = 8.
Hence, g{f(4)} = 8.
Therefore, f o g(1) = f{g(1)} = f(4) = 2 and g o f(4) = g{f(4)} = g(2) = 8.
问题2:使用下表,评估fog(3)和fof(1)。
x | f(x) | g(x) |
1 | 3 | 2 |
2 | 7 | 3 |
3 | 8 | 5 |
4 | 6 | 7 |
5 | 2 | 9 |
解决方案:
So, f o g(3) = f{g(3)} and f o f(1) = f{f(1)}.
We first evaluate the inner bracket then the outer bracket, using the values given in table.
For f{g(3)}, g(3) = 5. Now f(5) = 2.
Hence, f{g(3)} = 2.
Similarly, for f{f(1)}, f(1) = 3. Now f(3) = 8.
Hence, f{f(1)} = 8.
Therefore, f o g(3) = f{g(3)} = f(5) = 2 and f o f(1) = f{f(1)} = f(3) = 8.
评估复合函数:使用图
在这类问题中,我们将得到一个具有f(x)和g(x)曲线的图形,并且将要求我们找到f(x)和g(x)的组合,就像示例1中那样,找到雾(2)。
同样,我们将从内括号开始,因此我们必须找到g(2)。因此,我们将看到g(x)曲线。因此,在x = 2处,我们发现y = 3。现在我们需要找到f(3)。因此,我们将看到f(x)曲线。现在我们看x = 3,在那里我们找到y = 4。因此,我们得到了所需的解决方案,在示例1中为3。
问题1:使用下面的图形评估fog(2)。
解决方案:
So, f o g(2) = f{ g(2) }
From the graph of g(x), we need to find g(2),
when x = 2, y = 3, hence g(2) = 3
Now f o g(2) = f(g(2)) = f(3).
From the graph of f(x), we need to find f(3),
when x = 3, y = 4, hence f(3) = 4
Therefore, f o g(2) = f{g(2)} = f(3) = 4.
问题2:使用下图评估goh(5)。
解决方案:
So, g o h(5) = g{ h(5) }
From the graph of h(x), we need to find h(5),
When x = 5, y = -2, hence h(5) = -2
Now g o h(5) = g{h(5)} = g(-2).
From the graph of g(x), we need to find g(-2),
when x = -2, y = 3, hence g(-2) = 3
Therefore, g o h(5) = g{h(5)} = g(-2) = 3.