Question

The range of the real valued function \( f(x) = \log_3 (5 + 4x - x^2) \) is

• A. \( (0,2) \)
• B. \( [0,2] \)
• C. \( (-\infty,2] \)
• D. \( [-1,5] \)

Hint:

For logarithmic functions, determine the domain by ensuring the argument is positive. To find the range, analyze the behavior of the argument function and apply logarithm properties.

Answer 1

The Correct Option is C

We are given the function: \[ f(x) = \log_3 (5 + 4x - x^2). \] 

Step 1: Find the Domain of the Logarithm 
For the logarithmic function to be defined, the argument must be positive: \[ 5 + 4x - x^2>0. \] Rearrange the quadratic inequality: \[ -(x^2 - 4x - 5)>0. \] Factorizing the quadratic expression: \[ -(x-5)(x+1)>0. \] Solving \( (x-5)(x+1)<0 \) using the sign analysis method, the valid interval is: \[ -1<x<5. \] 

\Step 2: Find the Range of \( f(x) \) 
Since the quadratic expression \( 5 + 4x - x^2 \) is a downward-opening parabola, it attains a maximum value at its vertex. The vertex of the quadratic function \( g(x) = 5 + 4x - x^2 \) is given by: \[ x = \frac{-b}{2a} = \frac{-4}{-2} = 2. \] Substituting \( x = 2 \): \[ g(2) = 5 + 4(2) - (2)^2 = 5 + 8 - 4 = 9. \] Since the logarithm base is 3, we compute: \[ \log_3(9) = 2. \] As \( x \to -1^+ \) or \( x \to 5^- \), the argument \( 5 + 4x - x^2 \) approaches 0, meaning \( \log_3(5+4x-x^2) \) approaches \( -\infty \). Thus, the range of \( f(x) \) is: \[ (-\infty,2]. \]