Questions: ALEKS - 2024 Fall - College Algebra U15) Graphs and Functions Evaluating a piecewise-defined function Suppose that the function (h) is defined, for all real numbers, as follows. [ h(x)=left beginarrayll frac13 x^2-5 text if x neq-2 -3 text if x=-2 endarrayright. ] Find (h(-4), h(-2)), and (h(3)). [ beginarrayl h(-4)=square h(-2)=square h(3)=square endarray ]

Solution Steps

To evaluate the piecewise-defined function \( h(x) \), we need to determine which part of the function to use based on the value of \( x \). For \( h(-4) \) and \( h(3) \), since \( x \neq -2 \), we use the expression \(\frac{1}{3}x^2 - 5\). For \( h(-2) \), since \( x = -2 \), we use the constant value \(-3\).

Since \( -4 \neq -2 \), we use the expression for \( h(x) \) when \( x \neq -2 \): \[ h(-4) = \frac{1}{3}(-4)^2 - 5 = \frac{1}{3}(16) - 5 = \frac{16}{3} - 5 = \frac{16}{3} - \frac{15}{3} = \frac{1}{3} \]

Since \( x = -2 \), we use the defined value for \( h(x) \) at this point: \[ h(-2) = -3 \]

Since \( 3 \neq -2 \), we again use the expression for \( h(x) \) when \( x \neq -2 \): \[ h(3) = \frac{1}{3}(3)^2 - 5 = \frac{1}{3}(9) - 5 = 3 - 5 = -2 \]

Final Answer