Question

use the graphing tool to graph the functions $f(x) = \log_{4} x$ and $g(x) = \log_{0.25} x$. then identify the key features of ea drag each feature to the correct location on the table. features of f only features of both f and g features of g only domain of (0, ∞) range of (-∞, ∞) positive over the interval (0, 1) negative over the interval (0, 1) asymptote of x = 0 x-intercept of (1, 0)

Explanation:

Step1: Analyze \( f(x)=\log_4 x \) (base > 1, increasing)

• Domain: For any log function \( \log_b x \), domain is \( (0, \infty) \). Range: All real numbers, \( (-\infty, \infty) \). Asymptote: \( x = 0 \) (vertical). x - intercept: When \( y = 0 \), \( \log_4 x = 0 \Rightarrow x = 1 \), so \( (1, 0) \). Sign: For \( f(x)=\log_4 x \), when \( 0 < x < 1 \), \( \log_4 x < 0 \) (since \( \log_b x < 0 \) for \( 0 < x < 1 \) and \( b > 1 \)); when \( x > 1 \), \( \log_4 x > 0 \).

Step2: Analyze \( g(x)=\log_{0.25} x \) (base \( 0 < 0.25 < 1 \), decreasing)

• Domain: \( (0, \infty) \) (same as all log functions). Range: \( (-\infty, \infty) \). Asymptote: \( x = 0 \). x - intercept: \( \log_{0.25} x = 0 \Rightarrow x = 1 \), so \( (1, 0) \). Sign: For \( 0 < x < 1 \), \( \log_{0.25} x > 0 \) (since \( \log_b x > 0 \) for \( 0 < x < 1 \) and \( 0 < b < 1 \)); when \( x > 1 \), \( \log_{0.25} x < 0 \).

Step3: Categorize Features

• Features of \( f \) only: Positive over \( (1, \infty) \) (but from options, "negative over \( (0, 1) \)"? Wait, no: Wait \( f(x)=\log_4 x \) is negative when \( 0 < x < 1 \)? Wait no: \( \log_4 1 = 0 \), \( \log_4 (0.5)=\log_4 (1/2)= - \log_4 2 < 0 \), yes. Wait no, wait \( g(x)=\log_{0.25} x \): \( \log_{0.25} 0.5=\log_{1/4} (1/2)= (1/2) \) (since \( (1/4)^{1/2}=1/2 \)), so positive. So:
• Domain \( (0, \infty) \): Both (since all log functions have domain \( (0, \infty) \)).
• Range \( (-\infty, \infty) \): Both (log functions have range all reals).
• Positive over \( (0, 1) \): \( g \) only (since \( f \) is negative there, \( g \) is positive).
• Negative over \( (0, 1) \): \( f \) only (since \( f \) is negative, \( g \) is positive).
• Asymptote \( x = 0 \): Both (vertical asymptote at \( x = 0 \) for log functions).
• x - intercept \( (1, 0) \): Both (since \( \log_b 1 = 0 \) for any \( b > 0, b

eq 1 \)).

Wait, let's re - check:

• \( f(x)=\log_4 x \):
• Domain: \( (0, \infty) \)
• Range: \( (-\infty, \infty) \)
• Sign: \( f(x) < 0 \) when \( 0 < x < 1 \); \( f(x) > 0 \) when \( x > 1 \)
• Asymptote: \( x = 0 \)
• x - intercept: \( (1, 0) \)

• \( g(x)=\log_{0.25} x \):
• Domain: \( (0, \infty) \)
• Range: \( (-\infty, \infty) \)
• Sign: \( g(x) > 0 \) when \( 0 < x < 1 \); \( g(x) < 0 \) when \( x > 1 \)
• Asymptote: \( x = 0 \)
• x - intercept: \( (1, 0) \)

So:

• Features of \( f \) only: negative over the interval \( (0, 1) \)
• Features of both \( f \) and \( g \): domain \( (0, \infty) \), range \( (-\infty, \infty) \), asymptote \( x = 0 \), x - intercept \( (1, 0) \)
• Features of \( g \) only: positive over the interval \( (0, 1) \)

Answer:

• Features of \( f \) only: negative over the interval \( (0, 1) \)
• Features of both \( f \) and \( g \): domain of \( (0, \infty) \), range of \( (-\infty, \infty) \), asymptote of \( x = 0 \), x - intercept of \( (1, 0) \)
• Features of \( g \) only: positive over the interval \( (0, 1) \)

(To place in the table:

• Column "Features of \( f \) only": negative over the interval \( (0, 1) \)
• Column "Features of both \( f \) and \( g \)": domain of \( (0, \infty) \), range of \( (-\infty, \infty) \), asymptote of \( x = 0 \), x - intercept of \( (1, 0) \)
• Column "Features of \( g \) only": positive over the interval \( (0, 1) \))