52. If φ(x) = x² and ψ(x) = 2ˣ, then ψ(φ(x)) is:

(A) 2^x²

(B) x²

(C) 2^2x

(D) x^2x

Find ψ(φ(x)) When φ(x) = x² and ψ(x) = 2ˣ

Understanding the Given Function Composition Problem

This question is based on the concept of composition of functions. Two functions are given:

φ(x) = x²

and:

ψ(x) = 2ˣ

We need to determine:

ψ(φ(x))

The notation ψ(φ(x)) means that the output of the function φ is used as the input of the function ψ. Therefore, we first calculate φ(x) and then substitute that complete result into ψ.

Meaning of the Composition ψ(φ(x))

For two functions f and g, the composition:

f(g(x))

means that g(x) is evaluated first and its output is then substituted into the function f.

In the present question:

ψ(φ(x))

means:

First apply φ, then apply ψ

Therefore, the order of the functions is important. We do not simply multiply the two functions, and we do not apply them independently.

Step 1: Finding the Inner Function φ(x)

The inner function is:

φ(x) = x²

Therefore, the output of the first function is:

x²

This entire output will now become the input of the outer function ψ.

Step 2: Applying the Function ψ

The second function is:

ψ(x) = 2ˣ

This means that whatever input is given to ψ becomes the exponent of 2.

For example, if the input is represented by a general variable t, then:

ψ(t) = 2^t

In the composition ψ(φ(x)), the input to ψ is not simply x. The input is the complete expression:

φ(x) = x²

Therefore, substitute x² in place of the input of ψ.

Calculating ψ(φ(x))

We have:

ψ(x) = 2ˣ

Replacing the input x by φ(x) gives:

ψ(φ(x)) = 2^φ(x)

Since:

φ(x) = x²

we obtain:

ψ(φ(x)) = 2^x²

Therefore, the required composition is:

2^x²

Why the Entire Expression x² Becomes the Exponent

The function ψ(x) = 2ˣ assigns the input to the exponent of 2. If the input is 3, then ψ(3) = 2³. If the input is y, then ψ(y) = 2ʸ. Similarly, if the input is x², then:

ψ(x²) = 2^x²

Since the inner function produces x², the composition becomes:

ψ(φ(x)) = ψ(x²) = 2^x²

This is the direct application of the definition of function composition.

Verification With a Numerical Example

The result can be verified by choosing a simple value of x. Let:

x = 2

First, calculate the inner function:

φ(2) = 2² = 4

Now apply the outer function:

ψ(4) = 2⁴ = 16

Using the derived expression:

2^x²

and substituting x = 2:

2^2² = 2⁴ = 16

Both methods give the same result, confirming that:

ψ(φ(x)) = 2^x²

Difference Between ψ(φ(x)) and φ(ψ(x))

Function composition is generally not commutative. This means that changing the order of the functions can change the result.

The required composition is:

ψ(φ(x)) = ψ(x²) = 2^x²

However, if the order were reversed, we would get:

φ(ψ(x)) = φ(2ˣ)

Since φ(t) = t²:

φ(2ˣ) = (2ˣ)²

Using the exponent rule:

(a^m)ⁿ = a^mn

we obtain:

(2ˣ)² = 2^2x

Therefore:

ψ(φ(x)) = 2^x²

whereas:

φ(ψ(x)) = 2^2x

This distinction is especially important because 2^2x appears as one of the incorrect options.

Analysis of All the Given Options

Option (A): 2^x²

This option is correct. The inner function gives φ(x) = x². Substituting this output into ψ(x) = 2ˣ gives:

ψ(φ(x)) = ψ(x²) = 2^x²

Option (B): x²

This option is incorrect. The expression x² is only the output of the inner function φ(x). The question asks for ψ(φ(x)), so the function ψ must still be applied to this output.

Option (C): 2^2x

This option is incorrect for the required order of composition. The expression 2^2x is obtained when the functions are applied in the reverse order:

φ(ψ(x)) = (2ˣ)² = 2^2x

However, the question asks for ψ(φ(x)), not φ(ψ(x)).

Option (D): x^2x

This option is incorrect. Function composition does not involve multiplying the exponents or combining the functions in this manner. The output x² of the inner function must simply be substituted as the exponent in the outer function ψ.

Final Answer

Given:

φ(x) = x²

and:

ψ(x) = 2ˣ

we have:

ψ(φ(x)) = ψ(x²)

Therefore:

ψ(φ(x)) = 2^x²

Correct Option: (A) 2^x²