Math / Algebra

QuestionSelect Function Set 2, $f(x)=\sqrt{x}$ and $g(x)=x^{2}-4$ , and check the Values box. Using the $x$ -slider, set $x=3$ . Complete parts 1 through 3 below. Use the interactive figure to find your answer. Use the left and right arrow keys to move along a slider as needed. Click here to launch the interactive figure.
Part 1: Using the graph of g , evaluate $\mathrm{g}(3)$ . $g(3)=$ $\square$ 5 (Type an integer or decimal rounded to two decimal places as needed.) Part 2: Using the graph of $f$ and the value of $g(3)$ found in Part 1, evaluate $f(g(3))$ . $\mathrm{f}(\mathrm{g}(3))=$ $\square$ (Type an integer or decimal rounded to two decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We're given two functions, $f(x) = \sqrt{x}$ and $g(x) = x^2 - 4$ , and we need to find $g(3)$ and then use that result to find $f(g(3))$ . Watch out! Make sure to apply the functions in the correct order!
We need $g(3)$ first, then we plug that result into $f(x)$ .
Also, remember the order of operations when evaluating $g(3)$ .

STEP 2

1. Evaluate $g(3)$
2. Evaluate $f(g(3))$

STEP 3

We're given the function $g(x) = x^2 - 4$ .
To find $g(3)$ , we substitute $x = 3$ into the function: $g(3) = (3)^2 - 4$

STEP 4

Now, let's calculate the result!
First, we square the 3: $g(3) = 9 - 4$ Then, we subtract 4 from 9: $g(3) = 5$ So, $g(3) = \textbf{5}$ !

STEP 5

We just found that $g(3) = \textbf{5}$ .
Now, we need to find $f(g(3))$ , which is the same as finding $f(5)$ since $g(3)$ is 5.

STEP 6

Our function $f(x)$ is defined as $f(x) = \sqrt{x}$ .
We substitute our value of 5 for $x$ : $f(5) = \sqrt{5}$

STEP 7

Now, we calculate the square root of 5: $f(5) \approx 2.236$ Rounding to two decimal places, we get $\textbf{2.24}$ .

STEP 8

Part 1: $g(3) = 5$ Part 2: $f(g(3)) = 2.24$

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Studdy Solution

STEP 1

STEP 2

1. Evaluate $g(3)$
2. Evaluate $f(g(3))$

STEP 3

We're given the function $g(x) = x^2 - 4$ .
To find $g(3)$ , we substitute $x = 3$ into the function: $g(3) = (3)^2 - 4$

STEP 4

Now, let's calculate the result!
First, we square the 3: $g(3) = 9 - 4$ Then, we subtract 4 from 9: $g(3) = 5$ So, $g(3) = \textbf{5}$ !

STEP 5

We just found that $g(3) = \textbf{5}$ .
Now, we need to find $f(g(3))$ , which is the same as finding $f(5)$ since $g(3)$ is 5.

STEP 6

Our function $f(x)$ is defined as $f(x) = \sqrt{x}$ .
We substitute our value of 5 for $x$ : $f(5) = \sqrt{5}$

STEP 7

Now, we calculate the square root of 5: $f(5) \approx 2.236$ Rounding to two decimal places, we get $\textbf{2.24}$ .

STEP 8

Part 1: $g(3) = 5$ Part 2: $f(g(3)) = 2.24$

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Studdy Solution

STEP 1

STEP 2

1. Evaluate g(3)g(3)g(3)2. Evaluate f(g(3))f(g(3))f(g(3))

STEP 3

We're given the function g(x)=x2−4g(x) = x^2 - 4g(x)=x2−4.To find g(3)g(3)g(3), we **substitute** x=3x = 3x=3 into the function: g(3)=(3)2−4g(3) = (3)^2 - 4g(3)=(3)2−4

STEP 4

Now, let's **calculate** the result!First, we square the **3**: g(3)=9−4g(3) = 9 - 4g(3)=9−4 Then, we **subtract** 4 from 9: g(3)=5g(3) = 5g(3)=5 So, g(3)=5g(3) = \textbf{5}g(3)=5!

STEP 5

We just found that g(3)=5g(3) = \textbf{5}g(3)=5.Now, we need to find f(g(3))f(g(3))f(g(3)), which is the same as finding f(5)f(5)f(5) since g(3)g(3)g(3) is **5**.

STEP 6

Our function f(x)f(x)f(x) is defined as f(x)=xf(x) = \sqrt{x}f(x)=x​.We **substitute** our value of **5** for xxx: f(5)=5f(5) = \sqrt{5}f(5)=5​

STEP 7

Now, we **calculate** the square root of 5: f(5)≈2.236f(5) \approx 2.236f(5)≈2.236 Rounding to two decimal places, we get 2.24\textbf{2.24}2.24.

STEP 8

Part 1: g(3)=5g(3) = 5g(3)=5 Part 2: f(g(3))=2.24f(g(3)) = 2.24f(g(3))=2.24

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1. Evaluate $g(3)$
2. Evaluate $f(g(3))$

We're given the function $g(x) = x^2 - 4$ .
To find $g(3)$ , we substitute $x = 3$ into the function: $g(3) = (3)^2 - 4$

Now, let's calculate the result!
First, we square the 3: $g(3) = 9 - 4$ Then, we subtract 4 from 9: $g(3) = 5$ So, $g(3) = \textbf{5}$ !

We just found that $g(3) = \textbf{5}$ .
Now, we need to find $f(g(3))$ , which is the same as finding $f(5)$ since $g(3)$ is 5.

Our function $f(x)$ is defined as $f(x) = \sqrt{x}$ .
We substitute our value of 5 for $x$ : $f(5) = \sqrt{5}$

Now, we calculate the square root of 5: $f(5) \approx 2.236$ Rounding to two decimal places, we get $\textbf{2.24}$ .

Part 1: $g(3) = 5$ Part 2: $f(g(3)) = 2.24$