- Substitute \(x = -3\) into \(f(x) = x^2 + 7\) to get \(f(-3) = 16\).
- Substitute \(x = -2\) into \(h(x) = \frac{12}{x}\) to get \(h(-2) = -6\).
- Substitute \(x = 7\) into \(j(x) = 2x\) to get \(j(7) = 14\).
- A graph fails to represent a function if it intersects a vertical line at more than one point (Vertical Line Test).
- Final answers: \(f(-3) = \boxed{16}\), \(h(-2) = \boxed{-6}\), \(j(7) = \boxed{14}\).
Explanation
- Evaluating the Functions
To evaluate the given functions, we simply substitute the input value \(x\) into each function's formula and perform the indicated arithmetic.
- For the function \(f(x) = x^2 + 7\), we evaluate at \(x = -3\):
\[f(-3) = (-3)^2 + 7 = 9 + 7 = 16.\]
- For the function \(h(x) = \frac{12}{x}\), we evaluate at \(x = -2\):
\[h(-2) = \frac{12}{-2} = -6.\]
- For the function \(j(x) = 2x\), we evaluate at \(x = 7\):
\[j(7) = 2 \times 7 = 14.\]
- Identifying Non-Functional Graphs
A mathematical relation is defined as a function if every input (x-value) maps to exactly one output (y-value). Graphically, this is tested using the Vertical Line Test.
A graph does NOT represent a function if there exists at least one vertical line that intersects the graph at more than one point. This intersection implies that for a single x-value, there is more than one corresponding y-value, which violates the definition of a function. Conversely, if every possible vertical line crosses the graph at most once, the relation is indeed a function.
- Summary of Results
Based on the calculations above, the requested values are:
- \(f(-3) = 16\)
- \(h(-2) = -6\)
- \(j(7) = 14\)
And the condition for a graph not to represent a function is failing the vertical line test.
Examples
Understanding functions is crucial in real-life applications, such as tracking a budget over time. If \(t\) represents time in months and \(B(t)\) represents your bank balance, a function ensures that at any specific point in time, you have exactly one balance amount. If a graph of your spending were to fail the vertical line test (e.g., showing two different balances for the same date), it would indicate an error in accounting or an invalid model. Recognizing what constitutes a function helps us build logical and reliable mathematical models for everyday data.