Graph each equation. Give the domain and range. Identify any that are functions. (y)/(2)=√(1-(x^

step 1 So for this problem, we want to try to get this into a standard form to see what we're dealing w

Graph each equation. Give the domain and range. Identify any...

Key Concepts

Graphing Radical Equations

Graphing equations that involve radicals, particularly square roots, requires understanding how the basic graph of the square root function is transformed by operations like multiplication, division, or translation. This includes determining the vertex, axis of symmetry, and general shape of the graph while also considering restrictions from the domain. Graphing such equations aids in visualizing the relationship between variables.

Function Identification and the Vertical Line Test

Determining whether a plotted relation is a function involves applying the vertical line test—if any vertical line drawn across the graph intersects it at most once, the relation qualifies as a function. This concept is crucial in establishing the one-to-one correspondence needed between inputs (x-values) and outputs (y-values) and guarantees that the relation adheres to the definition of a function.

Domain Determination

This concept involves identifying all possible x-values that can be input into an equation so that the operations within the equation yield valid results. For equations with a square root, the expression under the square root must be greater than or equal to zero, leading to an inequality that restricts x. This process ensures that the equation is defined for the selected inputs.

Range Determination

Determining the range involves finding all the possible output values of the function or relation. For equations involving square roots, the function inherently produces nonnegative outputs (unless modified by additional operations), which helps in identifying the minimum and maximum values that y can take. The range is derived by considering how transformations, such as scaling or shifting, affect the basic square root function.

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