Question

1. let f be a decreasing function defined for x ≥ 0. the table gives values of f(x) at selected values of x. the function g is given by g(x) = log₂x.

(a) (i) the function h is defined by h(x) = (g ∘ f)(x) = g(f(x)). find the value of h(0), or indicate that it is not defined
(ii) find the value of f⁻¹(4), or indicate that it is not defined.

Explanation:

Part (A)(i)

Step1: Find \( f(0) \) from the table

From the table, when \( x = 0 \), \( f(0)=8 \).

Step2: Substitute \( f(0) \) into \( g(x) \)

The function \( g(x)=\log_{2}x \), so \( h(0)=g(f(0))=g(8) \).

Step3: Evaluate \( \log_{2}8 \)

We know that \( \log_{a}a^{b}=b \), and \( 8 = 2^{3} \), so \( \log_{2}8=\log_{2}2^{3}=3 \).

The inverse function \( f^{-1}(y) \) gives the \( x \) such that \( f(x)=y \). From the table, when \( f(x) = 4 \), the corresponding \( x \) value is \( 1 \). So we need to find \( x \) where \( f(x)=4 \), and from the table, when \( x = 1 \), \( f(1)=4 \). By the definition of inverse function, if \( f(x)=y \), then \( f^{-1}(y)=x \). So when \( y = 4 \), \( x = 1 \), which means \( f^{-1}(4)=1 \).

Answer:

\( h(0) = 3 \)

Part (A)(ii)