Given f(x)=4x2−3andg(x)=6−12x2 a. (f of g)(4)

Question

Given f(x)=4x2−3andg(x)=6−12x2  a. (f of g)(4)

Arnold Odonnell · Accepted Answer

Composite function:
A composite function is a function that depends on another function. This composite function is created by substituting one function into another function.
f(g(x)) is a composite function which is formed by substituting g(x) for x in f(x). The composite function f(g(x)) is also written as (f of g)(x) or fg(x). This can be read as “f of g of x".
The given two functions are

f (x) = 4 x^{2} - 3 and g (x) = 6 - \frac{1}{2} x^{2}

To obtain the composite function (f o g)(x), substitute g(x) for x in the function f (x).
The composite function (f o g)(x) is obtained as

141 - \frac{24}{x^{2}} + \frac{1}{x^{4}}

from the calculation given below:

(f \circ g) (x) = f (g (x))

= f b i g g (6 - \frac{1}{2 x^{2}} b i g g)

= 4 \times b i g {g (6 - \frac{1}{2 x^{2}} b i g g)}^{2} - 3

= 4 \times b i g g (36 - \frac{6}{x^{2}} + \frac{1}{4 x^{4}} b i g g) - 3

= 144 - \frac{24}{x^{2}} + \frac{1}{x^{4}} - 3

= 141 - \frac{24}{x^{2}} + \frac{1}{x^{4}}

Obtain the value of (f of g)(4).
The composite function (f of g)(x)is obtained as

= 141 - \frac{24}{x^{2}} + \frac{1}{x^{4}}

To obtain the value of (f o g)(4), substitute 4 in place of x in the composite function (f of g)(x)

= 141 - \frac{24}{x^{2}} + \frac{1}{x^{4}}

The value of (f o g)(4) is obtained as 139.5039 from the calculation given below:

(f \circ g) (x) = 141 - \frac{24}{x^{2}} + \frac{1}{x^{4}}

(f \circ g) (4) = 141 - \frac{24}{4^{2}} + \frac{1}{4^{4}}

= 141 - 1.5 + 0.0039

139.5039

Given f(x) = 4x^(2)-3 and g(x) = 6 - frac(1)(2)x^2 a. (f of g)(4)

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