Problem 49 If $f(x)=2 x+4, g(x)=x-1,$ and... [FREE SOLUTION]

Chapter 7: Problem 49

If $f(x)=2 x+4, g(x)=x-1,$ and $h(x)=x^{2},$ find each value. $$ h[f(-3)] $$

Short Answer

Expert verified

The value is 4.

Step by step solution

Evaluate f(-3)

First, we need to substitute into the function $f(x) = 2x + 4$. This will give us the value of $f(-3)$. Substitute $-3$ for $x$: \[f(-3) = 2(-3) + 4 = -6 + 4 = -2\].

Substitute f(-3) into h(x)

Now that we have $f(-3) = -2$, we substitute this value into $h(x) = x^2$. Replace $x$ with $-2$ in $h(x)$: \[h(-2) = (-2)^2 = 4\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation

Understanding function evaluation is like asking a machine to perform a specific task. Imagine a function as a recipe, and the variable is the ingredient you put in. Every time you have a specific value to plug into the function, you follow the steps it provides. Suppose you have a function $ f(x) = 2x + 4 $. If you're asked to evaluate $ f(-3) $, then everywhere you see an $ x $ in the recipe, you substitute $ -3 $. For our recipe, this means calculating $ 2(-3) + 4 $, which results in $ -6 + 4 = -2 $. Through function evaluation, you determine the output when you input a given value.

Substitution

Substitution in mathematics allows us to replace variables, like swapping puzzle pieces when you want to form a complete picture. When dealing with multiple functions, substitution is key. For our exercise, we first evaluated $ f(-3) $ to get $ -2 $. With substitution, this $ -2 $ now becomes the new value to work with. You then take this $ -2 $ and substitute it into another function, in this case, $ h(x) = x^2 $. In our puzzle, that means changing the $ x $ in $ h(x) $ to $ -2 $, leading us to calculate $ h(-2) = (-2)^2 = 4 $. This methodical replacement process helps simplify complex problems.

Quadratic Function

A quadratic function is a special type of function that forms a curve on a graph, known as a parabola. It's written in the form $ ax^2 + bx + c $. In our exercise, the function $ h(x) = x^2 $ is a simple quadratic function, where $ a = 1 $, and both $ b $ and $ c $ are zero. The beauty of quadratic functions is their symmetry and predictability. When they are evaluated with a specific number using substitution, as we did with $ -2 $ to find $ h(-2) = 4 $, they show how inputs relate to outputs in a way that's easy to graph. Understanding quadratics is crucial as they explain many real-world phenomena, like how objects fall under gravity.

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91影视

If \(f(x)=2 x+4, g(x)=x-1,\) and \(h(x)=x^{2},\) find each value. $$ h[f(-3)] $$

Short Answer

Step by step solution

Evaluate f(-3)

Substitute f(-3) into h(x)

Key Concepts

Function Evaluation

Substitution

Quadratic Function

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