QUESTIGN 6 Given: \( g(x)=\log _{\frac{1}{3}} x \) 6.1 Write down the domain of \( g \). 6.2 Calculate the \( x \)-intercept of \( g \). 6.3 Draw the graph of \( g^{-1} \), indicating the intercept(s) with the axes as well as the asymptote. 6.5 For which values of \( x \) will \( 1

Ask by Warner Williams. in South Africa

Feb 20,2025

The function \( g(x) = \log_{\frac{1}{3}} x \) is defined for all \( x > 0 \), so the domain of \( g \) is \( (0, \infty) \). Since logarithms are only defined for positive real numbers, any input value must satisfy this condition to yield a valid output. To find the \( x \)-intercept of \( g \), set \( g(x) = 0 \). This occurs when \( \log_{\frac{1}{3}} x = 0 \), which simplifies to \( x = 1 \). So, the \( x \)-intercept is the point \( (1, 0) \). This means that at \( x = 1 \), the function crosses the x-axis, which is a key point in understanding the behavior of the graph. The inverse of \( g(x) \) is \( g^{-1}(x) = \left(\frac{1}{3}\right)^{-x} = 3^{-x} \). The graph of this function will have an \( y \)-intercept at \( (0, 1) \), as \( g^{-1}(0) = 1 \), and will approach the x-axis (asymptote) as \( x \) goes to positive infinity. Thus, you would see a decreasing curve starting from the point (0,1) heading downwards to the right. For the inequality \( 1 < g^{-1}(x) < 3 \), we solve this within the context of the inverse function. First, set \( g^{-1}(x) = 1 \) to find \( x \) values where \( g^{-1}(x) \) equals 1, which gives \( x = 0 \). Then, set \( g^{-1}(x) = 3 \) to find where it equals 3, giving \( x = -1 \). Thus, this inequality is valid when \( -1 < x < 0 \), indicating specific values along the x-axis where the inverse function resides between these two outputs.