f(x) = 3x + 10, g(x) = x - 2. Find g(f(x)).
Understand the Problem
The question is asking to find the composition of two functions, specifically g(f(x)). We will compute f(x) first using the given formula, and then substitute that result into g(x).
Answer
$g(f(x)) = 3x + 8$
Answer for screen readers
The composition $g(f(x))$ is $3x + 8$.
Steps to Solve
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Calculate $f(x)$
We start with the function $f(x) = 3x + 10$.
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Substitute $f(x)$ into $g(x)$
Next, we need to find $g(f(x))$. We already computed $f(x)$, so substitute $f(x)$ into the function $g(x)$:
$$ g(f(x)) = g(3x + 10) $$
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Use the formula for $g(x)$
The function $g(x)$ is defined as $g(x) = x - 2$. Now we can replace $x$ in $g(x)$ with $f(x)$:
$$ g(3x + 10) = (3x + 10) - 2 $$
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Simplify the expression
Finally, we simplify the expression:
$$ g(3x + 10) = 3x + 10 - 2 = 3x + 8 $$
The composition $g(f(x))$ is $3x + 8$.
More Information
When composing functions, we take the output of the first function and input it into the second function. This shows how functions can work together to create new outcomes.
Tips
- Incorrect substitution: Forgetting to replace $x$ in $g(x)$ with the whole $f(x)$ expression.
- Not simplifying correctly: Failing to simplify after substitution, which can lead to errors in the final answer.
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