Question

explanation:
the output, y, of a relation, is the difference of three times the input, or 3x, and 5. so, ( y = 3x - 5 ).
substituting any x - value from the domain into the equation will result in exactly one value of y, so the relation is a function.
to find the range of the relation, given the domain, substitute the boundary point of the domain into the function equation:
when ( x = 2 ), ( y = 3(2)-5 = 6 - 5 = 1 ).
because substituting values of x that are greater than 2 will result in values of y that are greater than 1, the range of the relation is ( y>1 ).

question 2
the graph shows the relation ( y = x^{2} ). which of the following claims about this relation are true? use complete sentences to explain your reasoning.

• the relation is a function.
• the domain of the relation is all real numbers.
• the range of the relation is all real numbers.

Explanation:

1. The relation is a function: By the vertical line test, for any \( x \)-value (input), there is exactly one \( y \)-value (output) from \( y = x^{2} \). For example, if \( x = 2 \), \( y = 4 \); if \( x=- 2 \), \( y = 4 \). Each \( x \) maps to one \( y \), so it is a function.
2. The domain of the relation is all real numbers: The equation \( y=x^{2} \) is defined for any real number \( x \) (we can square any real number, positive, negative, or zero), so the domain is all real numbers.
3. The range of the relation is all real numbers: This is false. Since \( x^{2}\geq0 \) for all real \( x \) (the square of any real number is non - negative), the \( y \)-values (outputs) are always greater than or equal to 0. So the range is \( y\geq0 \), not all real numbers.

Answer:

• The claim "The relation is a function" is true. Reason: For every input \( x \) (any real number), the equation \( y = x^{2} \) gives exactly one output \( y \) (by the definition of a function and the vertical line test).
• The claim "The domain of the relation is all real numbers" is true. Reason: The expression \( x^{2} \) is defined for any real number \( x \) (we can square any real number), so all real numbers are valid inputs (domain).
• The claim "The range of the relation is all real numbers" is false. Reason: For \( y=x^{2} \), since the square of any real number \( x \) is non - negative (\( x^{2}\geq0 \) for all real \( x \)), the output \( y \) values are always greater than or equal to 0, not all real numbers.