Let `f(x)=x^(2)+2x-5` and `g(x)=5x+30`
Consider the following statements:
1 f[g(x)] is a polynomial of degree 3
2. g[g(x)] is a polynomial of degree 2 Which of the above is/are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to analyze the two statements regarding the functions \( f(x) = x^2 + 2x - 5 \) and \( g(x) = 5x + 30 \).

### Step 1: Evaluate \( f[g(x)] \)

We start by substituting \( g(x) \) into \( f(x) \):

\[ f[g(x)] = f[5x + 30] \]

Now, we substitute \( 5x + 30 \) into the function \( f(x) \):

\[ f[g(x)] = (5x + 30)^2 + 2(5x + 30) - 5 \]

### Step 2: Expand \( (5x + 30)^2 \)

Using the expansion formula \( (a + b)^2 = a^2 + 2ab + b^2 \):

\[ (5x + 30)^2 = 25x^2 + 300x + 900 \]

### Step 3: Expand \( 2(5x + 30) \)

Now we calculate \( 2(5x + 30) \):

\[ 2(5x + 30) = 10x + 60 \]

### Step 4: Combine all parts

Now we combine all parts together:

\[ f[g(x)] = 25x^2 + 300x + 900 + 10x + 60 - 5 \]

Combining like terms:

\[ f[g(x)] = 25x^2 + (300x + 10x) + (900 + 60 - 5) \]
\[ = 25x^2 + 310x + 955 \]

### Step 5: Determine the degree of \( f[g(x)] \)

The highest power of \( x \) in \( f[g(x)] \) is \( 2 \), hence the degree of \( f[g(x)] \) is \( 2 \).

### Step 6: Evaluate \( g[g(x)] \)

Next, we evaluate \( g[g(x)] \):

\[ g[g(x)] = g[5x + 30] \]

Substituting \( 5x + 30 \) into \( g(x) \):

\[ g[g(x)] = 5(5x + 30) + 30 \]

### Step 7: Simplify \( g[g(x)] \)

Calculating this gives:

\[ g[g(x)] = 25x + 150 + 30 \]
\[ = 25x + 180 \]

### Step 8: Determine the degree of \( g[g(x)] \)

The highest power of \( x \) in \( g[g(x)] \) is \( 1 \), hence the degree of \( g[g(x)] \) is \( 1 \).

### Conclusion

1. The first statement \( f[g(x)] \) is a polynomial of degree \( 3 \) is **incorrect** (it is of degree \( 2 \)).
2. The second statement \( g[g(x)] \) is a polynomial of degree \( 2 \) is **incorrect** (it is of degree \( 1 \)).

Thus, both statements are incorrect.

### Final Answer

Neither statement 1 nor statement 2 is correct.

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