Question

2
enter the correct answer in the box.
the function ( f(x) = 7^x + 1 ) is transformed to function ( g ) through a horizontal compression by a factor of ( \frac{1}{3} ). what is the equation of function ( g )?
substitute a numerical value for ( k ) into the function equation.
( g(x) = (7)^{kx} + 1 )

Explanation:

Step1: Recall horizontal compression rule

For a function \( y = f(x) \), a horizontal compression by a factor of \( \frac{1}{b} \) (where \( b>1 \)) transforms the function to \( y = f(bx) \).

Step2: Apply to given function

The original function is \( f(x)=7^{x}+1 \). A horizontal compression by a factor of \( \frac{1}{3} \) means we replace \( x \) with \( 3x \) in \( f(x) \) (since \( b = 3 \) for compression factor \( \frac{1}{3} \)). So \( g(x)=f(3x)=7^{3x}+1 \). Comparing with \( g(x)=7^{kx}+1 \), we see that \( k = 3 \).

Answer:

\( 3 \)