$3.$ Consider the following market demand function for good x: x$=12$m$-10$p$.-$SP$,+64,$ where A is advertising.

$($a$)$ Does the law of demand hold for this demand function?

$($b$)$ Are goods x and y substitutes or complements?

$($c$)$ Is good x a normal or an inferior good?

$($d$)$ Suppose that initially p$,=4,$ p$=10,$ m$=,$ and A$=10.$ Derive the equation for the inverse demand curve and graph it$.$

$($e$)$ Following from $($d$),$ determine the price elasticity of demand. If firms in the x$-$industry are contemplating decreasing the price of their product by $1\%,$ use the price elasticity of demand to determine whether the firms would benefit from this decrease. Is marginal revenue positive or negative currently?

$($f$)$ Using the values for prices, income, and advertising from $($d$),$ suppose that firms in the x$-$industry increase advertising expenditures by $1\%,$ what percentage change in demand for the product can they expect? Use an elasticity measure to answer this question.

$($g$)$ Suppose that a firm producing good x is deciding whether to lower its price by $1\%$ and keep its advertising expenditures unchanged. Its projections are that income growth in the region will be around $2\%.$ Is it able to use its elasticity measure from part $($e$)$ to determine the impact on demand from reducing the price?

$($h$)$ Determine if firms in the x industry should be very concerned if the price of good y falls by $1\%.$