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Match Input-Output Function Tables to Corresponding Algebraic Equations
Mathematics
High School Grade 9
Question Content
Match each function table (a, b, c, d) with its corresponding equation from the list: $y = \\frac{1}{x}$, $y = x^3$, $y = \\sqrt{x}$, $y = x^2$, $y = x$, $y = |x|$
Correct Answer
a. $y = x$; b. $y = |x|$; c. $y = x^3$; d. $y = x^2$
Detailed Solution Steps
1
Step 1: Analyze Table a. Test the input-output pairs with the equations: when input $x=-2$, output is $-2$; $x=1$, output is $1$. This matches $y=x$ because substituting any $x$ gives $y=x$ exactly.
2
Step 2: Analyze Table b. Test pairs: when $x=-2$, output is $2$; $x=-1$, output is $1$. This matches $y=|x|$ because the absolute value of a negative number is its positive counterpart, and non-negative inputs equal their outputs.
3
Step 3: Analyze Table c. Test pairs: when $x=-2$, output is $-8$ (since $(-2)^3=-8$); $x=3$, output is $27$ (since $3^3=27$). This matches $y=x^3$.
4
Step 4: Analyze Table d. Test pairs: when $x=-2$, output is $4$ (since $(-2)^2=4$); $x=3$, output is $9$ (since $3^2=9$). This matches $y=x^2$.
5
Step 5: Note that $y=\\frac{1}{x}$ and $y=\\sqrt{x}$ are not matched here: $y=\\frac{1}{x}$ cannot have $x=0$, and $y=\\sqrt{x}$ only accepts non-negative inputs, which do not align with any of the given tables.
Knowledge Points Involved
1
Linear Function ($y=x$)
A linear function where the output is exactly equal to the input. Its graph is a straight line passing through the origin with a slope of 1, and it is defined for all real numbers $x$.
2
Absolute Value Function ($y=|x|$)
A function that returns the non-negative magnitude of a real number $x$. For $x\\geq0$, $y=x$; for $x<0$, $y=-x$. Its graph is a V-shape with the vertex at the origin.
3
Cubic Function ($y=x^3$)
A polynomial function where the input is raised to the 3rd power. It preserves the sign of the input (negative inputs give negative outputs, positive inputs give positive outputs) and is defined for all real numbers $x$.
4
Quadratic Function ($y=x^2$)
A polynomial function where the input is squared. It always returns a non-negative output, as squaring any real number (positive or negative) results in a positive value, and $0^2=0$. Its graph is a parabola opening upwards with vertex at the origin.
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