Notation and Evaluation of Combined Functions
Use function notation to combine and evaluate functions: compute (f+g)(x), (fg)(x), (f/g)(x), and note domain restrictions where division or operations would be undefined.
Key Concepts
Property Functions can be combined using standard arithmetic operations to create entirely new functions. The notation tells you exactly which operation to perform on the outputs of the original functions: Addition: $(f + g)(x) = f(x) + g(x)$ Subtraction: $(f g)(x) = f(x) g(x)$ Multiplication: $(fg)(x) = f(x) \cdot g(x)$ Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$.
To evaluate a combined function at a specific number (like $x = 3$), you can either combine the algebraic expressions first and then plug in the number, or evaluate $f(3)$ and $g(3)$ separately and then combine their numerical results.
Examples Algebraic vs. Numerical: Let $f(x) = x + 5$ and $g(x) = 2x$. Find $(f + g)(3)$. Numerical method: $f(3) = 8$ and $g(3) = 6$. Add the results: $8 + 6 = 14$. Algebraic method: $(f + g)(x) = (x + 5) + 2x = 3x + 5$. Substitute 3: $3(3) + 5 = 14$. Division Domain Constraint: Let $f(x) = x + 4$ and $g(x) = x 2$. Find $(\frac{f}{g})(3)$. First, verify the denominator is not zero: $g(3) = 3 2 = 1$. Evaluate the numerator: $f(3) = 3 + 4 = 7$. The quotient is $\frac{7}{1} = 7$.
Common Questions
How do you add, subtract, multiply, and divide functions?
(f+g)(x)=f(x)+g(x), (f-g)(x)=f(x)-g(x), (fg)(x)=f(x)*g(x), and (f/g)(x)=f(x)/g(x) where g(x) is not zero. Perform the operation on the expressions for f and g, then simplify the result.
What domain restrictions apply to combined functions?
The domain of combined functions is the intersection of the domains of f and g. For (f/g)(x), additionally exclude any x-value where g(x)=0. Always state these restrictions explicitly because simplification may hide them in the resulting expression.
How do you evaluate (f+g)(3) when f(x)=2x and g(x)=x^2-1?
Compute f(3)=2(3)=6 and g(3)=3^2-1=8 separately. Then (f+g)(3)=f(3)+g(3)=6+8=14. Alternatively, form (f+g)(x)=2x+x^2-1 first and substitute x=3: 6+9-1=14.