CRE 4 - Composite Functions | Algebra - Functions & Graphs
f(x) = 2x + 3. Find f(f(2).
Answer: 17
Explanation :
Given, f(x) = 2x + 3.
∴ f(f(x)) = 2 × (2x + 3) + 3 = 4x + 6 + 3 = 4x + 9.
⇒ f(f(2)) = 4 × x + 9 = 17.
Hence, 17.
Workspace:
f(x) = 2x + 3 and g(x) = 5x – 4. Find f(g(1)).
Answer: 5
Explanation :
Given, f(x) = 2x + 3.
∴ f(g(x)) = 2g(x) + 3 = 2 × (5x - 4) + 3 = 10x – 5.
⇒ f(g(1)) =10 × 1 – 5 = 5.
Hence, 5.
Workspace:
If f(x + f(x)) = x2 – 3x + 5 and f(1) = 2, then find f(34).
- (a)
93
- (b)
39
- (c)
185
- (d)
190
- (e)
102
Answer: Option A
Explanation :
Given, f(x + f(x)) = x2 – 3x + 5 …(1)
putting x = 1, we get
f(1 + f(1)) = 12 – 3 × 1 + 5 = 3.
⇒ f(3) = 3
Now, substitute x = 3.
f(3 + f(3)) = 32– 3 × 3 + 5 = 5.
⇒ f(6) = 5
Now, substitute x = 6.
f(6 + f(6)) = 62– 3 × 6 + 5 = 23.
⇒ f(11) = 23.
Now, substitute x = 11.
f(11 + f(11)) = 112– 3 × 11 + 5 = 93.
⇒ f(34) = 93.
Hence, option (a).
Workspace:
If p(x) = 2x – 4, q(x) = x and r(x) = 1/x, then the formula for r(p(q(x))) is
- (a)
2x - 4
- (b)
1/(2x - 4)
- (c)
1/(2x - 4)2
- (d)
None of these
Answer: Option B
Explanation :
Given, p(x) = 2x – 4
∴ p(q(x)) = 2 × q(x) - 4 = 2(x) – 4 = 2x - 4
Given, r(x) = 1/x
∴ r(p(q(x))) = 1/p(q(x)) = 1/(2x - 4)
Hence, option (b).
Workspace:
A recursive formula is given by f(n) = f(f(n – 1)) + f(n – f(n – 1)), where n > 2 is an integer. If it is known that f(1) = 1 and f(2) = 1, find the value of f(4).
- (a)
1
- (b)
2
- (c)
3
- (d)
4
Answer: Option B
Explanation :
Given, f(n) = f(f(n – 1)) + f(n – f(n – 1)),
f(1) = 1 and f(2) = 1
putting n = 3, we get
f(3) = f(f(3 – 1)) + f(3 – f(3 – 1))
⇒ f(3) = f(f(2)) + f(3 – f(2))
⇒ f(3) = f(1) + f(3 – 1)
⇒ f(3) = 1 + f(2)
⇒ f(3) = 1 + 1 = 2
Now, putting n = 4, we get
f(4) = f(f(4 – 1)) + f(4 – f(4 – 1))
⇒ f(4) = f(f(3)) + f(4 – f(3))
⇒ f(4) = f(2) + f(4 – 2)
⇒ f(4) = 1 + f(2)
⇒ f(4) = 1 + 1 = 2
Hence, option (b).
Workspace:
If f(x) = 2x+3 and g (x) = (x – 3)/2, then what is the value of gof(x) - fog(x)?
- (a)
x
- (b)
0
- (c)
x2
- (d)
None of these
Answer: Option B
Explanation :
Given, f(x) = 2x + 3
∴ fog(x) = 2g(x) + 3 = 2((x - 3)/2) + 3 = x – 3 + 3 = x.
∴ gof(x) = (f(x) - 3)/2 = (2x + 3 - 3)/2 = x.
∴ gof(x) - fog(x) = x – x = 0.
Hence, option (b).
Workspace:
If f(x) = 2x + 3 and g (x) = (x – 3)/2, then what is the value of fofog(x)?
- (a)
f(x)
- (b)
0
- (c)
g(x)
- (d)
None of these
Answer: Option A
Explanation :
Given, f(x) = 2x + 3
∴ fog(x) = 2g(x) + 3 = 2((x - 3)/2) + 3 = x – 3 + 3 = x.
∴ fofog(x) = 2fog(x) + 3 = 2(x) + 3 = 2x + 3.
Hence, option (a).
Workspace:
If f(x) = |x| and g(x) = [x], then value of fog(-1/3) + gof(-1/3) is
where, |x| is the absolute function, and
[x] represents the greatest integer less than or equal to x.
- (a)
0
- (b)
1
- (c)
-1
- (d)
1/3
Answer: Option B
Explanation :
∵ fog(-1/3) = f[g(-1/3)] = f(-1) = 1
and gof(-1/3) = g[f(-1/3)] = g(1/3) = [1/3] = 0
∴ Required value = 1 + 0 = 1
Hence, option (b).
Workspace:
Find the value of + , if f(x) =
- (a)
2
- (b)
3
- (c)
4
- (d)
5
Answer: Option A
Explanation :
f(1/2) = = 2
f(f(1/2)) = f(2) = = 1
Also,
f(3) = =
f(f(3)) = f(1/2) = = 2
f(f(f(3))) = f(2) = = 1
∴ f(f(1/2)) + f(f(f(3))) = 1 + 1 = 2
Hence, option (a).
Workspace:
Feedback
Help us build a Free and Comprehensive Preparation portal for various competitive exams by providing us your valuable feedback about Apti4All and how it can be improved.