MTH100 Incharge Minahil Irshad by Ali Khan VU Nexus

Step 3: Find f (g (h(x)))

Now, substitute g (h(x) into f(x) :

f(g(h(x))) = f ( 𝑥2) = √ 1 𝑥2 = |𝑥 |

Final Result

Thus, the composite function (f o g o h) is: (f o g o h ) (x) = |𝑥|

Q2. Determine the domain and range of fog if f(x) = 1/x and g(x) = x + 1.

Answer

To determine the domain and range of the composite function , where

• f(x) = 𝑥 • g(x) = x+ 1

we will first compute (f o g ) (x) and then analyze its domain and range.

Step 1: Find (f o g ) (x)

The composite function is defined as: (f o g) (x) = f (g(x)) = f(x+1) Substituting g(x) into f(x) F(x+1) = 𝑥+1

Step 2: Determine the Domain

The domain of (f o g ) (x) = 𝑥+1 consists of all x values for which the function is defined. The function is undefined where the denominator is zero: X+1 ≠ 0 → x ≠ -1 Thus, the domain is:

Domain: (-∞,-1) U (-1 , ∞)

Step 3: Determine the Range

The range of f(x) = 𝑥 is all real numbers except zero. Since g(x) shifts the input but does not affect the overall behavior of f beyond avoiding division by zero, the range of (f o g) (x) = 𝑥+1 remains the same.

As x approaches -1 from the left, 𝑥+1 approaches -∞, and as x approaches -1 from the right, 𝑥+1approaches + ∞ Therefore, the function will never equal zero.

Thus, the range is: