QUESTION 5 KZNJ16 The equation \( f(x)=a^{x}, a>0, a \neq 1 \), passes through the point \( \left(-3 ; \frac{1}{8}\right) \). \( 5.1 \quad \) Determine the value of \( a \). \( 5.2 \quad \) Write down the equation of \( f^{-1} \) in the form \( y=\ldots \) \( 5.3 \quad \) Determine the value(s) of \( x \) for which \( f^{-1}(x)>-3 \).

To find the value of \( a \) for the function \( f(x) = a^x \) that passes through the point \( \left(-3; \frac{1}{8}\right) \): 5.1 Given the point \( f(-3) = \frac{1}{8} \), we have: \[ f(-3) = a^{-3} = \frac{1}{8} \implies a^{-3} = 2^{-3} \implies a = 2. \] So, the value of \( a \) is \( 2 \). 5.2 The inverse function \( f^{-1}(x) \) can be found by solving \( y = a^x \) for \( x \): \[ y = 2^x \Rightarrow x = \log_2(y). \] Thus, the equation of the inverse function is: \[ y = \log_2(x). \] 5.3 To find the values of \( x \) for which \( f^{-1}(x) > -3 \): \[ \log_2(x) > -3 \implies x > 2^{-3} = \frac{1}{8}. \] Therefore, for \( f^{-1}(x) > -3 \), \( x \) must be greater than \( \frac{1}{8} \).