Q. Find the domain and range of the function \(f(x) = 2^{2 – 3x}\).

Solution

1. Write the function and note its type.

   The function is \( f(x) = 2^{\,2 – 3x} \). This is an exponential function with base 2 (a positive number not equal to 1). Exponential expressions of the form \(a^{g(x)}\) with \(a>0\) are defined for every real value of the exponent \(g(x)\).

2. Find the domain.

   The exponent \(2 – 3x\) is a linear expression defined for all real numbers \(x\). Therefore the exponential is defined for every real \(x\).

   Domain: \( (-\infty, \infty) \).

3. Show that the function takes only positive values.

   For any real \(x\), \(2^{\,2 – 3x} > 0\) because a positive base raised to any real power is positive. Hence \(f(x)\) never equals 0 and never becomes negative.

4. Show that every positive real number is attained (so the range is all positive reals).

   Set \(y = 2^{\,2 – 3x}\) with \(y>0\). Take base-2 logarithms to solve for \(x\):

   \( \log_{2} y = 2 – 3x \).

   Rearrange to get \( x = \dfrac{2 – \log_{2} y}{3} \).

   For any chosen positive \(y\), \(\log_{2} y\) is a real number, so the formula gives a real \(x\) that produces that \(y\). Therefore every positive \(y\) occurs as an output of \(f\).

5. Conclude the range.

   From the previous steps, the outputs are exactly the positive real numbers.

   Range: \( (0, \infty) \).

Final answer:

Domain: \( (-\infty, \infty) \).

Range: \( (0, \infty) \).